Jürgen Klopp’s side trailed at the interval at Selhurst Park, with Luka Milivojevic’s penalty separating the teams during an opening 45 minutes that saw Sadio Mane have a goal disallowed and a spot-kick appeal rejected.
But they regrouped before returning for the second period in south London and were level within four minutes of the restart as Mane clipped home from James Milner’s assist.
Palace fought back and had chances of their own to claim three points but they were snatched by the visitors when Mohamed Salah struck for the 37th time this term in the closing moments.