Kevin Long’s own goal set Antonio Conte’s men on their way in the first half, but Ashley Barnes deflected Johann Berg Gudmundsson’s wayward shot to equalise for Burnley at Turf Moor.
Victor Moses’ right-footed shot into the bottom corner proved decisive and ensured the Blues, who trailed fourth-placed Tottenham by 10 points last week, narrowed the gap to five points.
Chelsea have the opportunity to increase the pressure on their London rivals when they travel to Swansea on 28 April.
Mauricio Pochettino’s team face Manchester United in the FA Cup semi-finals on Sunday and do not play in the league again until they meet Watford on 30 April.
Sean Dyche’s Burnley remain seventh and on course for the Europa League with an eight-point advantage over Leicester City in eighth.
With one eye on the semi-final, Chelsea’s strength in depth afforded Conte the luxury of making six changes to his starting XI without it impairing their performance.
They were accomplished going forward despite the absence of Cesc Fabregas and Willian, while Eden Hazard appeared only as a second-half substitute.
Chelsea fielded a two-man attack – in the form of Alvaro Morata and Olivier Giroud – for the first time in seven years.
While Chelsea’s finishing was profligate at times – particularly when Morata missed a one-on-one with Burnley goalkeeper Nick Pope after N’Golo Kante sent him clear – they were exemplary in their defensive work, restricting an obdurate Burnley to only two shots on target.