Midfielder Marco Alonso struck twice for the visitors who were in need of the points as much as the hosts.
Both sides played open football and Chelsea took the lead in the 33rd minute when Reece James crossed for Olivier Giroud to send his effort onto the crossbar.
Alonso was on hand to smash home the rebound.
Bournemouth bounced back with a headed goal from Jefferson Lerma nine minutes after the break.
And three minutes later, they were ahead when Joshua King tapped the ball home from close range after a superb ball in from the right by Jack Stacey.
Jolted into an all-out attack, Chelsea substitute Michy Batshuayi had a goal ruled out for offside moments after coming on, but they continued to bombard the Bournemouth box with set-pieces and crosses.
Their efforts paid off with five minutes of normal time left as Mason Mount fired a rocket which goalkeeper Aaron Ramsdale could only parry and Alonso reacted quickest to head home.
Chelsea stays in the fourth position on 45 points, while Bournemouth is still hovering above the drop zone on 27 points, two ahead of 17th-placed Aston Villa. (Reuters/NAN)