Having laboured in their 3-2 win over Derby County in the previous round, Chelsea found Bournemouth in resolute form at Stamford Bridge before substitute Hazard's 84th-minute winner.
Visiting goalkeeper Artur Boruc was in fine form throughout and Chelsea's luck looked to be out when Ruben Loftus-Cheek struck the woodwork early in the second half.
However, Hazard – who is now on 99 goals for the club in all competitions – took matters into his own hands, his strike from a tight angle taking a deflection off Charlie Daniels to find the net and book the home side a place in the last four.
Linked with a January switch to Stamford Bridge, Callum Wilson could have caught his reported suitors cold in the 16th minute but he fired wide at the culmination of a slick Bournemouth counter.
Eddie Howe's side had Boruc to thank when he denied both Olivier Giroud and then Willian, while Kepa Arrizabalaga had to be alert to tip Lys Mousset's effort wide.
Boruc was beaten by Loftus-Cheek's low strike after the interval, but the upright came to Bournemouth's rescue before Chelsea boss Maurizio Sarri turned to Hazard for inspiration.
The change paid off just as the prospect of penalties loomed over proceedings, Hazard exchanging passes with Pedro and Emerson Palmieri before drilling at goal, with Daniels' intervention only diverting the ball beyond his own keeper.
Hazard could have capped a brilliant cameo with two assists late on, yet Boruc twice made brilliant saves to prevent Emerson and also Giroud adding further gloss to the final scoreline.
Chelsea's Hazard reliance a concern
Sarri has plenty of quality at his disposal, yet without Hazard - who had picked up a knock in Chelsea's previous outing against Brighton and Hove Albion - the Blues lacked a cutting edge. While Chelsea can be happy with their progression to the last four, where they are joined by Manchester City, Burton Albion and Tottenham, Sarri will want his other star names to step up.
Hazard edges closer to Chelsea milestone
Hazard is now one shy of reaching a century of Chelsea goals in all competitions, and – given his sensational form this season – looks sure to reach the landmark soon. Even though Daniels' deflection proved key to his strike, the Belgian's craft and vision created the opportunity to get a shot away in the first place.
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