The teams had 36 shots on goal between them in a lively encounter, but Hazard's deflected effort six minutes from time proved to be the difference.
Chelsea now join Manchester City, Burton Albion and Tottenham Hotspur in the last four of the competition.
Find out how all of the action unfolded in West London courtesy of Sports Mole's minute-by-minute updates below.
CHELSEA SUBSTITUTES: Caballero, Zappacosta, Luiz, Kante, Pedro, Hazard, Hudson-Odoi
BOURNEMOUTH SUBSTITUTES: Travers, Mings, Surman, Pugh, Fraser, King, Defoe
CHELSEA XI: Arrizabalaga; Azpilicueta, Christensen, Rudiger, Palmieri; Kovacic, Fabregas, Barkley; Willian, Giroud, Loftus-Cheek
BOURNEMOUTH XI: Boruc; S.Cook, Simpson, Daniels; Ibe, Stanislas, Ake, Brooks, Rico; Wilson, Mousset
If you would like to read our on-the-whistle match report, you can do so by clicking here. Goodbye for now.