Results elsewhere had seen the champions slip down to fifth ahead of kickoff, 12 points adrift of leaders Manchester City, but Eden Hazard scored the only goal of the game to restore that gap back to nine and lift Antonio Conte's side above Arsenal into the final Champions League position.
Clear chances were few and far between during the opening exchanges, although Benik Afobe should have done better in the 17th minute when his pass to an unmarked Charlie Daniels forced the full-back too far wide and wasted a promising attacking platform.
Chelsea responded with a chance of their own, but two low crosses from Hazard in quick succession were dealt with by Asmir Begovic.
The former Chelsea keeper was at fault for the first gilt-edge chance of the contest, though, as his clearance went straight to Hazard, who in turn played the ball through for Alvaro Morata. The Spaniard only had the keeper to beat, but he put his finish wide of the target to spare Begovic's blushes.
Tiemoue Bakayoko then planted a header straight at the Bournemouth keeper before Chelsea did have the ball in the back of the net, only for Morata's strike to be disallowed for a tight offside call against Cesar Azpilicueta.
The visitors' pressure continued to grow, though, and Morata came close again with 10 minutes remaining of the first half when he was denied by a fine stop from Begovic.
Cesc Fabregas was the next to come close for Chelsea, but both of his efforts were blocked in a crowded penalty as Bournemouth continued to pack men behind the ball.
The Cherries brought on Jordon Ibe in place of Jermain Defoe at half time and began the second half in more adventurous fashion, but they fell behind within six minutes of the restart when Simon Francis failed to cut out Morata's ball to Hazard, who collected it and went on to beat Begovic with a powerful drive at the near post.
The hosts almost came up with a quick response, though, with Junior Stanislas capitalising on a defensive mix-up to find Ibe, who blasted his effort narrowly over the crossbar from the edge of the box.
Chelsea soon resumed control of proceedings, with Hazard and Pedro both threatening a second goal which would have all but killed Bournemouth's hopes off.
The hosts also suffered injury blows with Stanislas and Afobe limping off, but the latter's replacement Callum Wilson almost made an immediate impact on his first Premier League appearance of the season when his close-range effort was thwarted by a fine Azpilicueta block.
Fabregas's wait for his 50th Premier League goal ultimately went on, but the Spaniard had the chance to bring up the milestone with just seven minutes remaining when he collected a pass from Hazard only to scoop his finish over the crossbar from a tight angle.
Bournemouth had to wait until the 93rd minute for their first shot on target, and it almost earned them a point as Steve Cook blasted an effort towards goal which Thibaut Courtois was able to hold on to.
That proved to be the one and only scare for Chelsea, though, as they held out for a third consecutive victory across all competitions to get an important week off to a winning start with matches against Roma and Manchester United to come.
Bournemouth, meanwhile, remain in 19th place and still one point from safety having failed to improve on their worst start to a season since winning promotion to the Premier League.