Liverpool striker Mohamed Salah drew the penalty after getting tripped within the 18-yard box in the 45th minute by former Reds defender Mamadou Sahko.
Milner promptly converted from the spot to open his scoring account on the season and the 49th for his Premier League career.
Palace finished the match with 10 men when Aaron Wan-Bissaka was shown a red card in the 75th minute after a challenge on Salah, exiting the pitch on to a standing ovation from the Palace fans.
Sadio Mane sealed the win for Liverpool with a breakaway score in the 93rd minute.
Palace created several chances throughout the evening. Andros Townsend hit the bar with a left-footed shot from outside the box in the 23rd minute. Luka Milivojevic’s free kick to the top right corner was deflected by Liverpool goalkeeper Alisson in the 53rd minute.