The visitors were unable to convert a number of first-half chances, but quickly found the breakthrough following the restart thanks to Hazard.
Despite making further openings, Chelsea had to endure a nervy finish to the game, but they eventually closed out to claim all three points.
The first of Chelsea’s chances fell to Pedro. The winger drove at Bournemouth’s back line before unleashing a fierce strike high over Asmir Begovic’s goal.
Alvaro Morata had a golden opportunity to put Chelsea ahead mid-way through the first period. Begovic's loose clearance fell to Hazard, who played in the striker, but he fired wide from close range.
The Blues had a goal questionably ruled out on 28 minutes. Chelsea worked a short corner to David Luiz on the edge of the box, whose shot was deflected before falling to Azpilicueta in an offside position. Despite the flag being raised, Begovic still parried the defender's strike to Morata, who tucked home before play was halted.
Morata was again on the end of a decent chance for Chelsea just after the half-hour mark. Hazard's superb first touch fell to Morata, who brilliantly turned before seeing a strike well saved by the arm of the Bournemouth goalkeeper.
The travelling fans did not have to wait long after the break for their team to open the scoring. Hazard got on the end of Morata's lofted ball over the top and fired in at the near post.
Bournemouth almost had the perfect response on 56 minutes when substitute Jordon Ibe picked up the ball on the edge of the box and unleashed a strike on goal. Antonio Rudiger, however, got there just in time to deflect over the bar and save his side.
Hazard, who at this point was almost guaranteed to pick up the game’s man-of-the-match award, couldn’t seal the points after being found by Pedro in the box, while the Spaniard failed to beat Begovic following a scintillating one-touch Chelsea move.
Bournemouth did have one final chance to earn a point, but Steve Cook’s attempt was comfortably saved by Thibaut Courtois in stoppage time.
With that, Chelsea closed out the game to move back into the top four, while Bournemouth remain in 19th position in the Premier League table with just seven points from ten games.