Liverpool dominated possession with over 66% in the first half and enjoyed the majority of shots on goal.
The goal marked Mane’s eighth straight goal against Palace and his brace on Sunday took his total goals against the side to 12.
Palace sought an equaliser in the second half but none materialised as Liverpool took advantage with a second Mane goal.
Salah, who went into the day tied on 22 goals with Harry Kane in the goalscoring charts, selflessly passed the ball off to Mane on the edge of the box who fired it home on 74 minutes.
The result means Palace manager Roy Hodgson finished his tenure with the club on a loss.
Hodgson took charge of Palace in September 2017 and recorded their best-ever Premier League points total of 49 in 2018-19.