Chris Wood scored the only goal of the game when keeping his composure to beat Vicente Guaita from Jay Rodriguez's flick.
Johann Berg Gudmundsson missed the opportunity to kill the game off when his first time shot cannoned back off the bar, leaving the Clarets to fight tooth and nail in a bid to protect their lead.
Palace applied the pressure with Nick Pope making vital saves to keep out Michy Batshuayi and substitute Christian Benteke.
Here are the ratings...
1. Nick Pope 8
Kept Palace out with a number of big saves. Denied Clyne and Townsend in the first half, spread himself well to prevent Batshuayi from equalising in the second half and then burst half his line to make a vital stop from substitute Benteke.
2. Matt Lowton 6
The full back had a testing evening up against Palace's summer signing Eze, who got beyond the full back too easily at times. The pace of the away side's attacks left him vulnerable at times, but he battled his way to a clean sheet.
3. James Tarkowski 7
Prevented both Ayew and Batshuayi from laying a glove on the Clarets in the first half and did everything within his power to prevent Palace from breaking through when things got a bit more desperate after the break. A strong display.
4. Ben Mee 7
The captain won everything in the air and on the ground. Stuck tight to Palace's forwards, ensuring they had their backs to goal, and picked up some good positions to clear when the pressure mounted after the break. Leading by example.