The visitors took the lead mid way through the first half when the Burnley goalkeeper Nick Pope pushed a cross into Kevin Long and the ball ricocheted off the defender into his own net.
Chelsea striker Alvaro Morata should have made it 2-0 on the hour mark. He burst clear on goal, but, with just Pope to beat, side-footed wide of the target.
Four minutes later, Chelsea paid the price for the lack of precision.
Joey Gudmundsson's long-range effort was heading wide until it deflected off Ashley Barnes to wrong-foot the Chelsea goalkeeper Thibaut Courtois.
But after 69 minutes Victor Moses restored Chelsea's lead with a shot into the right of the goal. The 2017 Premier League champions held on to register back-to-back wins in the Premier League for the first time in 2018.
Burnley's first defeat in seven games dealt a blow to their hopes of qualifying for a European competition for the first time since 1967.
But Sean Dyche, the Burnley manager, remained upbeat about the progress of a side that were once considered relegation fooder.
"I am really pleased with how we are going about our business this season," said Dyche. "We're closer than where we have been in the past where we have been with these sides.
"We are not them. We are not close to them yet but we have moved forward."