The home side had an excellent first half after Kepa Arrizabalaga saved Erling Haaland’s penalty, before Evanilson gave the Cherries the advantage at half time.
But Pep Guardiola’s second half substitutes made the difference, Nico O’Reilly coming off the bench to set up Haaland for the equaliser four minutes after the resumption.
And O’Reilly was again involved to slip in fellow substitute Omar Marmoush to add the Citizens’ second shortly after the hour mark.
The Cherries were unable to find an equaliser to take the game to extra time and they were eliminated from the FA Cup at the quarterfinal stage.
Andoni Iraola made three changes to his team which lost to Brentford before the international break.
Marcos Senesi returned from injury to start at centre back in place of the suspended Dean Huijsen, while Milos Kerkez was replaced by Julio Soler and David Brooks came in for Marcus Tavernier.
_(Image: Richard Crease)_ Pep Guardiola named a strong side including Erling Haaland, Kevin De Bruyne and Phil Foden.
The Cherries made an energetic start to the tie, causing problems for City with their press but unable to fashion any clear-cut early chances.
The best early opening came for the visitors, Erling Haaland heading a Matheus Nunes cross wide, before the Manchester side were awarded a penalty after 11 minutes.
But Cherries keeper Kepa Arrizabalaga guessed the right way and kept Haaland’s spot kick out, before the Norwegian put a good opportunity over the bar when one-on-one with the Spaniard.
_(Image: Richard Crease)_ Antoine Semenyo shot over the bar after he won the ball on the left and drove into the box.
The Cherries grabbed the lead after 21 minutes, Lewis Cook finding Brooks overlapping on the right, the Welshman clipping the ball to the arriving Justin Kluivert at the far post.
The Dutchman got to the ball first and poked the ball goalwards past Ederson, and Evanilson was there to smash home from less than a yard.
Semenyo flashed a cross across goal from the left, before a Nunes cross/shot was claimed by Kepa.
_(Image: Richard Crease)_ The Cherries went close to a second, Brooks clipping a free kick into the box and Senesi was millimetres away from the decisive touch.
De Bruyne curled an effort over the bar from just outside the box in the sole minute of stoppage time at the end of the first half as the Cherries took their one-goal advantage into the break.
It didn’t take long for the visitors to draw themselves level after the break.
De Bruyne found the onrushing halftime substitute Nico O’Reilly on the City left, and he put it on a plate for Haaland to finish at the back post four minutes into the second period.
Andoni Iraola went to his bench after 57 minutes, introducing Dango Ouattara and James Hill for Brooks and Soler.
_(Image: Richard Crease)_ And City’s substitutes would make the difference again shortly after the hour mark.
O’Reilly won the ball after Semenyo went to ground to challenge, and he slipped in Marmoush who had replaced Haaland just two minutes previous, his shot finding the back of the net after going underneath the diving Kepa.
Senesi cleared a Marmoush header off the line, before Iraola brought Alex Scott and Adam Smith on for Evanilson and Ryan Christie.
Ilkay Gundogan struck the post with a shot from just outside the box, before Senesi was replaced by Daniel Jebbison with six minutes to play as the Cherries sought a late equaliser.
Kepa made a point blank save as the game became hectic, the Cherries unable to find the equaliser to take the game to an additional 30 minutes, and they were knocked out of the cup.
**AFC Bournemouth:** 13, Kepa; 20, J. Soler (J. Hill 57’); 5, M. Senesi (D. Jebbison 85’); 27, I. Zabarnyi; 4, L. Cook (c)(booked, 70’); 12, T. Adams; 10, R. Christie (booked, 62’)(A. Smith 74’); 24, A. Semenyo; 19, J. Kluivert (booked, 77’); 7, D. Brooks (D. Ouattara 57’); 9, Evanilson (A. Scott 74’).
**Manchester City:** 31, Ederson; 24, J. Gvardiol; 3, Rúben Dias; 45, A. Khusanov (booked 37')(N. O'Reilly 46'); 27, Matheus Nunes (booked, 86’); 19, İ. Gündoğan (J. McAtee 83’); 8, M. Kovačić (N. Gonzalez 83’); 20, Bernardo Silva; 17, K. De Bruyne (c); 9, E. Haaland (O. Marmoush 61’); 47, P. Foden (J. Grealish 69’).