In a hard-fought encounter, Andros Townsend gave the Eagles the lead on 34 minutes when he finished off a counter-attack by converting Wilfried Zaha’s low cross from close range.
Mohamed Salah equalised immediately after the restart, touching in Virgil van Dijk’s deflected strike, before Roberto Firmino put them in front moments after.
James Tomkins headed in a leveller from a corner just after the hour; however, Salah claimed his second when he tapped in from on the line after Julian Speroni had misjudged James Milner’s cross.
Milner was shown a second yellow card with a minute of normal time left to play following a challenge on Zaha, but Palace’s hopes of salvaging a point were ended when Sadio Mane netted in stoppage time.