Johann Berg Gudmundsson fired the Clarets in front after the Palace defence failed to deal with an Erik Pieters cross and the Icelandic winger created space for himself before drilling a left-foot shot into the far corner.
Burnley doubled their advantage from a Dwight McNeil corner into a packed box, with Palace keeper Vicente Guaita unable to come out and claim the ball, allowing Jay Rodriguez to head home his first Premier League goal of the season.
Matt Lowton’s first ever Burnley goal at this level made it 3-0 after the break, the full-back powering forward from the half-way line, finding Rodriguez inside and then volleying home the return pass.
Sean Dyche’s side headed to Selhurst Park as the lowest scorers in the league but struck twice within the opening 10 minutes.
There was a sour end to a fine performance from the visitors, though, when central defender Ben Mee was stretchered off with a serious concussion head injury.
Burnley are now on 26 points, 11 above 18th-placed Fulham who occupy the highest of the relegation spots. Palace are in 13th place on 29 points.