Chris Richards, Jordan Ayew and Jean-Philippe Mateta scored the goals for Palace, who have moved into 14th position in the Premier League table, eight points clear of the relegation zone.
Palace had a half-chance inside the first minute through Jefferson Lerma, but the midfielder, who recently scored a spectacular goal against Chelsea, missed the target from outside the box.
The Eagles were dominating the contest, and Odsonne Edouard came close to meeting a dangerous cross from Ayew in the 18th minute, but the striker just could not make contact.
Burnley goalkeeper James Trafford made an outstanding save to keep out a close-range effort from Joachim Andersen in the 24th minute, but the referee's whistle blew soon after for offside.
Trafford was again called into action just before the 30-minute mark to keep out a header from Edouard, with Burnley holding on, and the away side's task became even tougher in the 34th minute when Josh Brownhill was handed a straight red card for pulling Lerma to the ground on the edge of the box.
From the resulting free kick, Edouard's free kick just grazed the top of the crossbar, and the two teams were level at the interval despite Palace's domination for long spells.
The hosts also had the first opportunity of the second period through Lerma, but the midfielder's strike was saved by Trafford.
Adam Wharton was next to come close for Palace, just missing the post with a curling effort, with the home side continuing to put the pressure on Burnley.
David Datro Fofana headed one wide of the post for the visitors in the period which followed, but Palace were straight back on the front foot and in search of the game's opening goal.
The breakthrough came in the 68th minute, with Richards on the scoresheet, heading a brilliant delivery from Ayew into the bottom corner of the net.
Palace then doubled their advantage in the 71st minute, with Ayew tapping in at the far post after smart work from Matheus Franca in a wide area.
The Eagles were then awarded a penalty in the 76th minute following a foul from Vitinho on Franca, and Mateta stepped up to convert the spot kick to secure all three points for Glasner's side.
Burnley thought that they had scored in the latter stages through Fofana, but the effort was chalked off following a lengthy VAR check, which summed up the away side's afternoon.
Palace will continue their league campaign away to Tottenham Hotspur next Saturday, while Burnley will welcome Bournemouth to Turf Moor next Sunday.