Harry Wilson briefly gave Bournemouth hope by reducing the deficit to 2-1 with a stunning 25-yard free-kick in first-half stoppage time. But Silva laid off the ball for Aguero's second in the 64th minute to make sure of the victory for City, which saw its 15-match winning run in the league ended by a draw against Tottenham.
City dominated possession on the south coast but also survived some shaky moments. With 13 minutes on the clock, City keeper Ederson escaped with a yellow card after racing out of his box and clattering Callum Wilson wide on the Bournemouth left.
Sections of the home support called for more severe punishment from referee Andre Marriner and their frustration was compounded two minutes later. Neat interplay by Spanish midfielder Silva and left-back Oleksandr Zinchenko led to Kevin De Bruyne scuffing an attempted shot into the path of Aguero and the Argentinian took a touch before calmly slotting into the bottom-left corner.
De Bruyne's inadvertent assist was his 50th in 123 Premier League appearances, breaking Mesut Ozil's record of 141 for the fewest matches required to reach the milestone. The hosts had a further setback in the 37th when Charlie Daniels, who recently returned from a long-term knee injury, was carried off on a stretcher with a similar problem and replaced by Liverpool loanee Wilson.
City continued to wobble defensively and Nicolas Otamendi almost gifted the hosts an equalizer a minute later. The Argentine opted to chest down the ball inside his own box, allowing Adam Smith to nip in and see his close-range effort unwittingly thwarted by the head of Ederson.
It was more difficult to fault the away side's attacking play and they went further ahead two minutes before the break. After collecting a pass from Bernardo Silva, David Silva slipped in Sterling, allowing the England forward to poke home his 11th goal in 10 outings against the Cherries.
City had won each of the previous eight Premier League meetings between the clubs, registering an aggregate score of 25-3. Any hopes they had of easing to another emphatic victory were dealt a blow deep into first-half stoppage time.
Callum Wilson was fouled by French defender Laporte 25 yards from goal and winger Harry Wilson, who opened his Bournemouth account at Aston Villa last week, produced an unstoppable left-footed free-kick that flew into the top-right corner off the angle of post and crossbar.
The second half began in frenetic fashion with England striker Callum Wilson denied by Ederson after getting goal-side of Laporte, before De Bruyne curled narrowly wide at the other end. City was then denied what appeared to a clear spot-kick when Jefferson Lerma stood on the foot of star man Silva. A VAR review surprisingly ruled no penalty but City did not have to wait long to restore their two-goal advantage.
Bournemouth conceded possession cheaply from a Smith throw-in, allowing David Silva to jink into the box before Aguero applied the final touch with a stabbed finish. The Cherries pushed for a way back into the game but were unable to prevent a first defeat of the campaign as City's dominance of this fixture continued.
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