Goals from Bernardo Silva, Raheem Sterling and Ilkay Gundogan sealed the Cherries' fate after a spirited first-half display in which they subdued City's attack for large parts. During the second period City started to dominate, but Cherries boss Eddie Howe believes his side were still in the game until it was 3-1 and bemoaned the defensive errors leading to the goals.
He told Sky Sports "We had moments and unfortunately for us the three goals we conceded were out of context with the game and were scrappy. They weren't goals where they cut through us with beautiful football and you kick yourself a little bit for that.
"I think the team have to do better and I think the second goal is a key moment in the game. The third one is as well because at 2-1 we are still very much in the game," he added.
City's second goal scored by Sterling is what Howe was making reference to - a shot from Danilo was inexplicably parried back out into the centre of the box by Asmir Begovic and the Englishman was fastest to react ahead of a number of players to prod the ball home.
Howe was also left frustrated that his side were not awarded a penalty during the second half, believing that playing away from home did not help Bournemouth's cause.
"It is one of those situations where as the away team, with no noise to disturb the referee, it is an easy one to wave away. I think possibly it could have been a penalty."