Chelsea were ahead on 32 minutes through Hazard, who has belatedly hit form in recent months after a disappointing campaign for last year's PFA Player of the Year.
The result means Liverpool must win at West Brom and hope Southampton lose or draw at home to Crystal Palace to finish seventh and have a chance of qualifying for the Europa League next season.
However, seventh will only qualify for the Europa League if Manchester United win in the FA Cup final against Palace and finish in the top six. Liverpool can of course qualify for the Champions League if they beat Sevilla in the Europa League final on May 18.
Liverpool had the better of the game early on, with Baba Rahman's block six yards out denying Philippe Coutinho a sure goal after Alberto Moreno had pulled the ball back to him on the penalty spot.
