Second-placed City broke the deadlock inside six minutes as Spanish midfielder Silva curled a sublime free kick into the top corner for his second set-piece goal in three league appearances.
City doubled their advantage when Jesus slalomed past two defenders to score from close range, with Silva collecting his 10th assist of the league campaign.
Bournemouth avoided further damage in the second half when the VAR overturned a City penalty awarded for a tackle by defender Steve Cook on Brazilian striker Jesus.
The duo went close to an equaliser in the closing stages but City held on to collect all three points and extend their winning streak at home to six games.
"It was difficult but our opponent played really well, they were really good and they pushed us a lot," City manager Pep Guardiola said.
"Their needs are higher than ours but we played our game."
City have already secured a second-placed finish behind title winners Liverpool, while Bournemouth remain 18th and three points adrift of the safety zone.
Eddie Howe's Bournemouth recorded 14 attempts on goal but poor finishing left them in 18th place, three points adrift of the safety zone with two games remaining.
"We gave it everything tonight and real credit to the players to keep on going until the final whistle," Howe said.
"When you create those moments - and you don't get many against this quality of team - you need to take them.
"We need to match the performance level here and maybe surpass that against Southampton, that is all we should focus on now."
City have secured second place behind champions Liverpool and next face Arsenal on Saturday in the FA Cup semi-finals.