But in the last minute of the first half Mohamed Salah was brought down by Mamadou Sakho and Milner converted his 13th successful Premier League spot-kick.
Alisson saved from Luka Milivojevic and Wilfried Zaha and Palace’s search for an equaliser became harder when Aaron Wan-Bissaka was shown a straight red card for fouling Salah when through on goal.
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The Reds took advantage of the extra man in stoppage time when Salah assisted Mane for his third goal of the season.
Second-placed Liverpool are the sixth club to win each of their first two fixtures. The Eagles stay on three points in 10th.
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