Everton were sharper from the start and won a penalty when Calvert-Lewin was tripped by Liverpool keeper Alisson
Their defeat by Crystal Palace on Saturday re-ignited the title race but Chelsea responded with a gritty display to keep second-placed Tottenham Hotspur -- 3-1 winners at Swansea City -- safely at arm's length with eight games left.
Sergio Aguero levelled with a tap-in after 26 minutes following a poor clearance by keeper Thibaut Courtois but Hazard restored Chelsea's lead against the run of play in the 35th with a rebound after his weak penalty was saved by Caballero.
City skipper Vincent Kompany, starting a league match for the first time since November, grazed the top of Chelsea's crossbar early in the second half.
The victory gave Chelsea 72 points to Tottenham's 65.