The match saw both teams end with 10 players on the field as Manchester City's Mateo Kovacic and Bournemouth's Lewis Cook received red cards in the 67th and 73rd minutes respectively.
Omar Marmoush scored the first goal for Manchester City in the 14th minute with a powerful shot from 30 yards out that curved into the top-left corner of the net.
Bernardo Silva doubled the lead for Manchester City in the 38th minute after Ilkay Gundogan passed the ball back to him, allowing Silva to slot it into the near post.
Nico Gonzalez scored the third goal for Manchester City in the 89th minute with a run from deep that ended with a shot past goalkeeper Kepa Arrizabalaga.
Bournemouth's Daniel Jebbison scored a late consolation goal just before the final whistle.
The match was an emotional night for Manchester City captain Kevin De Bruyne, who played his final game at Etihad Stadium after 10 years with the team. Although he had a close-range miss in the first half, he was given a standing ovation when he was substituted out in the 69th minute.
Midfielder Rodri made his return to action after an eight-month absence as a late substitute.