The visitors grew in confidence as the game went on and almost equalised through Dominic Solanke, but his goal-bound effort was blocked expertly by Nicolas Otamendi.
City continued to dominate possession and Silva then turned provider just before the interval, setting up Gabriel Jesus, who jinxed past multiple opponents before arrowing an effort into the bottom corner from inside the box.
Josh King had the ball in the net for the Cherries before half-time but he was judged to be offside after a lengthy VAR review.
VAR was in the spotlight once again midway through the second period as Jesus tumbled inside the 18-yard box. Referee Lee Mason pointed to the spot, before overturning his decision after consulting with VAR.
The Cherries battled well throughout and pulled a goal back through David Brooks with just two minutes remaining in the match.
Guardiola's men weathered the late storm to claim all three points!
View the best images from the match via the gallery below.