Antonio Conte’s team went into a deserved lead on 20 minutes at Turf Moor. Moses burst down the right and his cross went in off Kevin Long for an own goal.
Nine minutes after Alvaro Morata had missed a great chance to double the lead, Burnley were level.
Johann Berg Gudmundsson’s 64th-minute shot to the near post was inadvertently deflected into the other corner by Ashley Barnes.
Chelsea soon restored their lead through Moses. Emerson Palmeri’s cross found the wing-back at the far post and he fired in to close the gap on Tottenham Hotspur, in fourth.
The defeat harmed Burnley’s hopes of moving into the sixth place occupied by Arsenal, who are two points ahead with a match in hand.
Burnley: 22 Apr v Stoke (A)
Chelsea: 28 Apr v Swansea (A)