Chelsea had buzzed round the Bournemouth goal for much of the game but it was not until substitute Hazard came on with half an hour to go that they looked creative and decisive enough to break through.
Chelsea had piled on the pressure from the start and there were good chances for Olivier Giroud, Ross Barkley, who forced a diving save from Artur Boruc and Willian, who surged forward from the halfway line after 30 minutes only for Boruc to block his rising shot.
Bournemouth looked dangerous on the break and Callum Wilson ran through on goal in the 17th minute but pulled his shot wide.
A scampering run from Jordon Ibe opened up the Chelsea defence 20 minutes later and Chelsea keeper Kepa Arrizabalaga had to fly to his left to keep out Lys Mousset's curling shot.
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Ruben Loftus-Cheek looked certain to break the deadlock early in the second half after rounding two players but his shot flew inches past Boruc's post.
Chelsea brought on first Pedro then Hazard and it was a combination of the Spaniard and Belgium's Hazard that made the goal.
"The difference was the changes that we made," Chelsea's assistant boss Gianfranco Zola told reporters.
"Of course we have a lot of games and players like Eden (Hazard) we need to keep fit. Even in 20 minutes he can make a big difference."
Another player who caught Zola's eye was England midfielder Loftus-Cheek.
"I was impressed with him. He has done well as a wide player, but he showed a lot of sacrifice as a midfielder," Zola said.
"He was getting tired but was disciplined... He is a good boy and works very hard."
It was the first home game since alleged racist taunts against Manchester City's Raheem Sterling prompted an outcry and Chelsea banned four fans.
Chelsea Chairman Bruce Buck took a page in the programme to condemn racism and the atmosphere in the ground was lively but appeared good humoured with no added security in evidence.
Chelsea will play Tottenham Hotspur in the semi-finals.
(Reporting by Clare Lovell,; Editing by Toby Davis and Ed Osmond)
