Hazard had been an injury doubt ahead of the game, but the Belgian came off the bench to net the decisive strike in an entertaining encounter in West London.
The Blues join Manchester City, Burton Albion and Tottenham Hotspur in the last four, while Bournemouth return to Premier League action having suffered seven defeats from eight matches in all competitions.
Chelsea went straight onto the front foot with Willian and Ross Barkley both forcing saves out of Artur Boruc during the opening stages of the game, but it was not long before Bournemouth began to display their ability in attack.
Ruben Loftus-Cheek was making a positive impression out on the flank and the Englishman's pass was able to find the run of Willian who only had Boruc to beat, but the stand-in stopper remained big to make a timely block.
Jordon Ibe was proving to be a threat on the right flank and a powerful run from the former Liverpool man gave Lys Mousset the chance to shoot from distance, but Kepa Arrizabalaga was equal to the Frenchman's effort from distance.
Boruc kept out Willian's long-range attempt just before the break, while Giroud, Loftus-Cheek and Barkley all failed to make the most of opportunities in and around the box.
Both teams struggled to get on top after the restart, but Loftus-Cheek came as close as anyone to an opening goal with the Chelsea academy graduate hitting the outside of the post with a low drive after powering towards the penalty area.
Three yellow cards in quick succession disrupted the flow of the match, and it left both sides struggling to create anything of note as the game progressed into the final quarter.
Hazard had been introduced as a substitute on the hour mark but the Belgian failed to make an impact until being denied by Boruc in the 79th minute, with the playmaker getting a shot away after a rapid counter from the home side.
However, Hazard had his goal soon afterwards. The attacker was given the chance to shoot from eight yards out after collecting a clever drag-back from Pedro, and his effort took a huge deflection on its way to beating Boruc and finding the far bottom corner.
Emerson Palmieri and Giroud were both denied by further outstanding saves from Boruc during four minutes of added-on time, but Chelsea will be satisfied with a narrow win which moves them to within one tie of a place in February's final.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Christensen (Luiz 81'), Rudiger, Palmieri; Kovacic, Fabregas, Barkley (Hazard 61'); Willian (Pedro 55'), Giroud, Loftus-Cheek
BOURNEMOUTH (3-5-2): Boruc; S.Cook, Simpson, Daniels; Ibe, Stanislas (Fraser 63'), Ake, Brooks (Defoe 88'), Rico; Wilson, Mousset (King 73')