The Belgian, Chelsea's main threat throughout, punished poor home defending six minutes after the break when he lashed a shot past Asmir Begovic for his third goal in his last three games for the London side.
Second-bottom Bournemouth were always in contention, though, and Chelsea manager Antonio Conte looked relieved as the final whistle sounded on the south coast.
It was a crucial victory for Chelsea who remained fourth with 19 points, nine behind leaders Manchester City.
"I think we deserved to win because we played good football," Conte said.
"To keep the result to only 1-0 can be dangerous. I think today that if I have to find a negative aspect, we could be more clinical."
Alvaro Morata had a goal disallowed and was denied by a great save from Begovic in a first half in which Bournemouth coped reasonably comfortably with the visitors' attack.
Chelsea went ahead in the 51st minute though when home defender Simon Francis failed to cut out a ball over the top and Hazard took advantage to beat Begovic at his near post.
Bournemouth responded tenaciously but lacked a cutting edge and their only shot on target came in stoppage time when Steve Cook drilled a shot straight at Thibaut Courtois.
(Reporting by Martyn Herman, editing by Ed Osmond)
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