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.
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Palace applied the pressure with Nick Pope making vital saves to keep out Michy Batshuayi and substitute Christian Benteke.