Palace were knocking on the door for an equaliser after the break but saw their chances of snatching a point hit by Aaron Wan-Bissaka's red card 15 minutes from the end.
The young full-back brought down Salah on a Liverpool break, and it was another lightning counter which Mane rounded off which sealed the Reds victory in stoppage time.
Relive how the action unfolded East London courtesy of Sports Mole's minute-by-minute updates below.
Subs: Guaita, Ward, Kelly, Kouyate, Meyer, Sorlorth, Ayew
Subs: Mignolet, Matip, Moreno, Henderson, Lallana, Shaqiri, Sturridge