Having laboured in a 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.
⏱ @ChelseaFC is also into the last four after a single, scrappy goal from @hazardeden10 handed the Blues a 1-0 victory over @afcbournemouth. Watch LIVE and catch-up on beIN SPORTS CONNECT https://t.co/EaHIo88tNn #CarabaoCup #CHEBOU pic.twitter.com/F15nh7BUBl— beIN SPORTS (@beINSPORTS_AUS) December 19, 2018
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, substitute 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.