Milner opened the scoring with a penalty just before the break after Mohamed Salah was fouled while attempting to wriggle away from Mamadou Sakho.
And Mane wrapped the three points up in the 93rd minute, the No.10 racing onto Salah’s pass and rounding Wayne Hennessey after the Reds had absorbed a late flurry from Palace - who had seen Aaron Wan-Bissaka sent off with a quarter of an hour left for tripping the Egyptian as he bore down on goal.
The hosts went close to opening the scoring when Andros Townsend struck the crossbar with a long-range effort, and Alisson Becker was forced to tip Luka Milivojevic’s free-kick wide as they pressed for an equaliser before Wan-Bissaka’s dismissal.