Mane levelled four minutes after the break, but, having already been booked for diving, was later given a huge let off by referee Kevin Friend when he deliberately handled the ball just outside his own penalty area.
Palace remain two points above the drop zone.
Liverpool suffered a huge blow in the 70th minute when Adam Lallana had to be taken off just six minutes after entering as a substitute. His finish from James Milner's driven cross was sharp, but from Palace's perspective, all too simple.
Roy Hodgson faces some difficult selection decisions ahead of Palace's game on Saturday
Salah proved to be the match-winner once again, however, as the Egypt global fired a low Andrew Robertson cross past Hennessey in the 84th minute to break Palace hearts late on.
Jurgen Klopp immediately moved to replace Mane with Alex Oxlade-Chamberlain. Alexander-Arnold had a torrid start to the game; the 19-year-old is an exceptional prospect but was clearly targeted by Palace, who looked to expose him with early releases to Zaha.
The Eagles became frustrated and began to tire in the in the final 15 minutes, which eventually led to Liverpool snatching all three points on 84 minutes.
Maybe Lallana only picked up a slight knock and will be back in time for Liverpool's Champions League clash against Man City on Wednesday. The rest of the first half was not much better with Liverpool looking frustrated and poor in possession. That season, the Ivorian scored 37 goals in 44 games.
With only nine goals in 29 outings, few tears were shed when Christian Benteke was allowed to leave Merseyside after one underwhelming season. After an terrible start to the season and the sacking of Frank de Boer, Roy Hodgson has turned things around, but they're not clear of danger yet.
Former Piston Grant Hill selected for Hall of Fame
In his career , he averaged 16.7 points per game, 6 rebounds per game, 4.1 assists and had a 48.3 field goal percentage. The future Hall of Fame member made 10 All-Star appearances and six All-NBA teams during his career .