Bournemouth held their own during the opening three quarters of the match in West London, but clinical strikes from two of Chelsea's flair players made the difference.
Both teams went into the contest having found goals easy to come by during their first three fixtures, but clear-cut chances were few and far between during a lively opening.
Chelsea remained in the ascendancy as the first half progressed with Willian dragging a shot marginally wide of Asmir Begovic's post, but the best opportunity of the opening 45 minutes went to the visitors.
Summer signing Diego Rico delivered a pinpoint cross from the left flank, but after Callum Wilson had timed his run perfectly to get in behind the Chelsea defence, the forward sent a half-volley harmlessly over the bar.
That near-miss only encouraged Bournemouth, who persisted with attempting to hit Chelsea on the break, but the struggle for chances continued as the teams went into the interval.
Chelsea returned for the restart with the intention of applying constant pressure to the Bournemouth defence, but while that materialised, Sarri's players were still left frustrated by the visitors.
Marcos Alonso saw a first-time effort comfortably saved by Begovic and, moments later, former Chelsea man Nathan Ake squandered the chance to put his team in front at the other end when firing over from no more than three yards out.
Sarri acted by introducing both Olivier Giroud and Pedro, and it took less than 10 minutes for the pair to combine to give Chelsea what was a deserved advantage.
The duo played a sharp one-two on the edge of the penalty area, and after Pedro had taken a touch to set himself from 18 yards, the Spaniard's resulting shot took a slight deflection on its way into the bottom corner of the net.
The goal forced Bournemouth to push forward in search of an equaliser but despite a change in approach, Chelsea continued to control the match and a second goal arrived five minutes from the end of the contest
Alonso picked up the ball on the left-hand side before threading a pass through to Hazard, who wasted little time before drilling a low effort into the far corner of Begovic's net.
Bournemouth almost halved the deficit through Ryan Fraser - who curled the ball agonisingly wide of the far post from distance - but Chelsea were full value for their win and will be delighted with the start to their season heading into the international break.
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger, Luiz, Alonso; Jorginho, Kovacic (Loftus-Cheek 79'), Kante; Willian (Pedro 65'), Hazard, Morata (Giroud 61')
BOURNEMOUTH (3-4-3): Begovic; Francis, Ake, S.Cook; A.Smith (Ibe 80'), Gosling, Lerma (L.Cook 88'), Rico; King (Mousset 80'), Fraser, Wilson