At 3-1 down, Pep Guardiola's team had one foot out of the competition but ended up mounting an epic comeback led by a lethal Sergio Aguero.
The Argentine neutralized Radamel Falcao's double with two of his own and helped City to a 5-3 victory that the home support will never forget.
City took the lead through Raheem Sterling but were then playing catch-up as Monaco drew level thanks to Radamel Falcao's header and went ahead with a powerful finish from 18-year-old Kylian Mbappe five minutes before halftime.
South American poachers Sergio Aguero for City and Falcao, who also missed a penalty, then traded blows with Aguero equalizing for 2-2 and then 3-3 after Falcao's exquisite chip gave Monaco the lead for the second time at The Etihad.
City, however, then pulled clear as Stones tapped in at the far post and Sane rolled the ball into an empty net to give City a two-goal advantage ahead of the return on March 15.
City manager Pep Guardiola had called Monaco "killers" in the box before the game and while they showed just why they are European football's most deadly assassins, they were embarrassingly fragile at times at the back.
They were behind after 26 minutes when Sterling tapped home after Sane dazzled down the left and teed up the England winger.
Monaco were handed a route back into the match after 32 minutes following an error by City's Willy Caballero, the first of the night's goalkeeping blunders.
He gave the ball away with a poor clearance and the visitors worked it to Fabinho whose devilish cross was headed home by former Manchester United and Chelsea striker Falcao, who has found his best form again since returning to France.
City were denied a penalty by Spanish referee Antonio Mateu Lahoz when Aguero appeared to be upended by keeper Danijel Subasic before Monaco grabbed the lead through Mbappe.
The 18-year-old was released through the middle and lashed his finish into the top corner past Caballero.
The visitors should have extended their advantage soon after the restart but Falcao's tame penalty was saved after Nicolas Otamendi had been penalized for catching the Colombian as he tried to convert Benjamin Mendy's cross.
City swiftly made Monaco pay as Aguero ended a six-game goal-drought thanks to a Subasic howler. Sterling slid the ball through and the Argentine fired straight at the keeper but it squirmed out of his hands into the net after 58 minutes.
Parity was short-lived, however, as Falcao turned City Defender Stones before lifting an exquisite chip over the stranded Caballero to make it 3-2 to the principality club.
Aguero levelled once more when he volleyed home from 12 meters after a corner and City took the lead when Kevin De Bruyne's set-piece was turned in by Stones at the far post.
The final twist in one of the most entertaining encounters came in the competition's recent history came when Aguero rolled the ball into the path of Sane who tapped into an empty net.
5 - Manchester City: Caballero; Sagna, Stones, Otamendi, Fernandinho (Article 62); Yaya Toure, Sane, Silva, De Bruyne, Sterling (Jesus Navas 89), Aguero (Fernando 89).
3 - Monaco: Subasic; Sidibe, Glik, Raggi, Mendy; Silva (Joao Moutinho 85), Fabinho, Bakayoko (Dirar 88), Lemar; Mbappe (Germain 79), Falcao.
Goals: 1-0, m.26: Sterling. 1-1, m.32: Falcao. 1-2, m.40: Mbappe. 2-2, m.58: Aguero. 2-3, m.61: Falcao. 3-3, m.71: Aguero. 4-3, m.77: Stones. 5-3, m.82: Sane.
referee: Antonio Mateu Lahoz (ESP).
Yellow cards: Aguero (m.35), Fernandinho (m.40), Otamendi (m.49), Zabaleta (m.73) and Glik (m.8), Sidibe (m.25), Bakayoko (m.55), Falcao (m.66), Bernardo Silva (m.76) and Fabinho (m.81).
Griezmann's reaction to Gameiro's change: No, he's the best, he's the best!
When Diego Simeone decided to change Kevin Gameiro in the game against Bayer Leverkusen, the cameras of Antena3 captured the spontaneous reaction of Antoine Griezmann.
His compatriot had produced a great performance, but at 3-2, Simeone brought on Thomas Partey.
"No, no, he's the best, he's the best!" Greizmann exclaimed.
After the match, he was asked about that reaction.
"The coach knows better," he continued.
"Gameiro was helping us a lot with his diagonals, but the coach was right because we caught the fourth.
Simeone: The angrier players are about substitutions, the happier I am
Atletico Madrid produced a stunning attacking performance in Leverkusen, especially in the first half, and Diego Simeone stood by his decision to take off his impressive forwards Kevin Gameiro and Antoine Griezmann in the 71st and 78th minutes, while he was not upset at their apparent frustration.
In fact, the Argentine coach explained that he was pleased to see such passion from his players as they were replaced, with Thomas Partey, Angel Correa and Fernando Torres all coming to seal the win.
"The angrier players are about substitutions, the happier I am," Simeone insisted, before explaining the logic behind his hooking of Gameiro.
"They were pushing when our lead was at 3-2 and everything seemed more dangerous than it had done before.
"Gameiro was having one of the best matches he has had since arrived and [Griezmann] was doing his thing, but my goal was to introduce fresh players.
"I think it worked out because [Torres] scored one and Correa had a chance too."
The coach did, however, admit that Los Rojiblancos' best spell came before the break in a first half which could have ended with a larger lead than the 2-0 they took to the dressing room.
"What I take away from this match is our play, which was, particularly in the first half, tactically perfect," he told reporters at his post-match press conference.
"We intelligently took the game to where we could cause them problems.
"The result could have been greater and their goalkeeper had a great game.
"With their goal, they got themselves into the match more in the second half.
"We were able to do what we'd discussed beforehand."
Saul Niguez conjured up the opener with a long-range rocket, something Simeone knows is capable of.
"Saul has a fantastic shot and we knew that, by playing in that area, he could make life difficult for Bayer with his left foot," he explained.
"His goal was extraordinary."
While he was delighted at the fighting play going forward, the Argentine did admit that he would have liked a little better showing in defense.
"It has been 21 years since Atletico Madrid scored four goals in the Champions League," he said.
"We need to try to finish with fewer goals conceded, which is vital in this tournament."
Finally, he refused to claim that the tie was over, pointing to the fact that anything could happen in the return leg on March 15.
Attack is the best form of defense for City boss Guardiola
Facing Europe's most potent strikeforce would cause some managers to man the barricades but for Manchester City boss Pep Guardiola there was only one option in Tuesday's Champions League clash with Monaco - "attack, attack, attack".
As it turned out, it proved to be a remarkably effective strategy as City came out on top in an epic 5-3 tussle, clinging to the coat-tails of their opponents before drawing clear in the closing stages of their last 16, first -leg tie.
There was perhaps a risk in going toe-to-toe with a team that has scored 76 goals in 26 domestic French league games and had already soared past the century in all markets this season, but the gamble paid off handsomely.
Goals from Raheem Sterling and a double from Sergio Aguero kept City in the encounter, before late efforts from John Stones and Leroy Sane saw them pull clear ahead of the second leg in the principality on March 15.
"We think in attack. Attack, attack, attack," Guardiola told BT Sport.
"Monaco score maybe 80 goals in one season and attack with a lot of people. They are physically strong and are a top, top team. That's why they're top of the league. That's why we take a lot of credit with this result . "
The match will go down as one of the greatest Champions League encounters of the modern era and for City, a side with little European pedigree, will help build a bank of experience they can use to help them develop, Guardiola said.
The Spaniard's last two clubs - Barcelona and Bayern Munich - have won 10 European Cups between them, but his latest charges are relative novices among Europe's elite, having only last season got past the round of 16 for the first time.
"For our future as a club, to live this kind of experience will help us a lot," Guardiola said.
"Monaco has more history than us in the competition and you need this kind of experience to learn and improve. Of course anything can happen in Monaco and we have to score goals."
Neither side's defense covered themselves in glory during a match that was riddled with errors at the back, while both teams scored due to goalkeeping howlers.
Monaco striker Radamel Falcao also missed a penalty in a match where the overall goal tally could have reached double figures, such was the open-ended nature of the game.
Despite finishing on the losing side, however, there were few regrets for Monaco coach Leonardo Jardim, who is confident his side can still go through over the two legs.
"The first thing I'll say is that maybe this was one of the most incredible matches in the Champions League this season," he told reporters.
"It's great for the fans as I'm sure they were happy to watch such quality attacks and eight goals. I think Monaco played really well and I've congratulated the players.
"We've still got 90 minutes to play and it's far from over."
The best Gameiro of the season
Atletico Madrid have been waiting a long time for Kevin Gameiro to have a night like the one he had at the Bay Arena on Tuesday night. In a competition like the Champions League, the Frenchman showed that he has quality to be the main striker for the Rojiblancos. If he had been questioned during the season, he won’t be again.
First, he assisted Griezmann for the second goal, then added the penalty that he himself scored.
Considering his and Atleti's spot-kick woes this season, he was brave to step up to take it but got his reward.
Thus, Gameiro announced the fourth goal penalty kick for el Athletic en la presente campaña después de que haya disfrutado de diez ocasiones el club rojiblanco. Torres se topó con el dûruero ante el Celta, Griezmann también falló contra Leganés, Valencia y Bavaria y Gabi tampoco being actuated against the conjunctive that.
It was Gameiro's fourth successful penalty for Atletico this season after seeing Torres hit the crossbar from 12 yards against Celta, Griezmann also missing against Leganes, Valencia and Bayern, and Gabi also unsuccessful.