The match was six minutes away from going to penalties when Pedro’s back-heel gave Hazard a chance, with his shot deflecting off Charlie Daniels and past Artur Boruc.
Bournemouth, who were aiming to reach the semi-final of a major cup competition for the first time in their history, had chances but Callum Wilson shot just wide and Lys Mousset had an effort saved by Kepa Arrizabalaga.
Chelsea join 2018 winners Manchester City, Tottenham and League One side Burton Albion in the last four.
Bournemouth, with keeper Boruc joining the attack, had a chance from a free-kick in the 94th minute but shots from Ryan Fraser and Wilson were blocked as the hosts held on.
It means Chelsea boss Maurizio Sarri remains on course for a cup final in his first season in English football.
His team, fourth in the Premier League and eight points behind leaders Liverpool, had the majority of the chances but Boruc saved efforts from Ross Barkley, Olivier Giroud and Willian in the first half.
The impressive Ruben Loftus-Cheek hit the foot of the post with a fine strike from outside the penalty area before Hazard replaced Barkley in the 61st minute.
The Belgium international shot over the bar and had another attempt from 30 yards well saved by Boruc, before he linked up with Pedro for the game’s only goal – his 10th for Chelsea in all competitions this season.