Sean Dyche's side headed to Selhurst Park as the lowest scorers in the league and with just five goals on the road, but struck twice within the opening 10 minutes.
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 crowded out and 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 halfway line, finding Rodriguez inside and then volleying home the return pass.
"VAR wasn't checking for offside, they were checking if it actually happened," joked Lowton, whose last goal in the Premier League came for Aston Villa almost eight years ago.
There was a sour end to a fine performance from the visitors, though, when central defender Ben Mee was taken off on a stretcher with a head injury, although Dyche reported after the game that the player was "fine."
The Burnley manager had been questioned over his decision to play a largely second-string XI in their FA Cup exit to Championship side Bournemouth on Tuesday and felt the win justified his call.
"I felt with injuries and the fact that players give a lot, it's important to make those changes and today it paid us back," he said.
"I thought the energy levels of the side were terrific, the tempo both in and out of possession and the effective play that I talk about many, many times and I thought it was all on show in a very complete performance," added Dyche.
Palace manager Roy Hodgson could not disguise his disappointment with his team.
"It was pretty obvious that it was a performance we can't be in any way happy with or proud of. We were beaten by the better team," he said.
"I am afraid it is one of those defeats that you have to take on the chin but there won't be many of us going home thinking we did the best job we could do," said the former England boss.
Burnley are now on 26 points, 11 above 18th-placed Fulham who occupy the highest of the relegation spots. Palace are three points ahead of the Clarets in 13th place.