Substitute Christopher Nkunku netted a late goal to snatch a 1-0 away win for Chelsea despite a dominant attacking display from Bournemouth.
The Cherries’ £40m record signing Evanilson had earlier missed an excellent chance to score his first Premier League goal when he had a poor first-half penalty parried away by Robert Sanchez.
Bournemouth constantly tormented Chelsea’s defence, capitalising on their often lacklustre distribution and getting in behind the back line.
But the hosts did not make the most of having the bulk of the chances and were made to pay by Nkunku, who prodded the ball past Mark Travers after a through-ball from debutant Jadon Sancho.
It was a hard-fought and aggressive encounter on the south coast, with referee Anthony Taylor handing out a Premier League record 14 yellow cards.
Bournemouth had more final-third opportunities but did not manage to find a way to get ahead as both Marcus Tavernier and Ryan Christie had efforts denied by the woodwork.
Chelsea, who move up to seventh, will play West Ham at 12:30 BST next Saturday in their next Premier League game, while 11th-placed Bournemouth will visit Liverpool at 15:00 BST later that day.