A fantastic first half saw Andoni Iraola's side lead 1-0 at halftime, but a poor second half allowed City to gain control of the tie, and once they took the lead shortly after the hour mark, they never looked like spurning it.
The atmosphere inside the Vitality Stadium was bouncing for the first 45 minutes as the Cherries got on the front foot, boosted by a Kepa Arrizabalaga penalty save.
Evanilson handed Bournemouth the lead after 21 minutes, which they took into the break.
But after the interval, City gained control and equalised four minutes after the resumption, Erling Haaland atoning for his missed penalty by finishing from a pinpoint cross from substitute Nico O'Reilly.
And fellow substitute Omar Marmoush gave the visitors the lead after 63 minutes, shooting home under the diving Kepa, after being slipped through by O'Reilly.
The Cherries were not able to find an equaliser to extend the tie to extra time, and they were knocked out of the FA Cup at the quarterfinal stage.
Freelance photographer Richard Crease was at the game for the Daily Echo.
See a gallery of his pictures above.