Joe Ledley gave Palace a second-half lead against the run of play after Liverpool had dominated the opening 45 minutes.
James Milner received a red card shortly after Palace’s goal to leave Liverpool a difficult task to get back into the game.
But Roberto Firmino capitalised on Alex McCarthy’s mistake to level before substitute Christian Benteke converted an added-time penalty to seal a 2-1 victory.
Jurgen Klopp’s side are six points behind fourth-placed Manchester City following their win at Selhurst Park.
Former Liverpool defender Carragher believes the Reds’s slender win can inspire a late push for a top-four finish.
“They are six points behind Man City,” Carragher told Sky Sports. “It is going to be very difficult to get top four.
“It gives Liverpool players hope that the next eight or nine games won’t peter out.
“There is a light at the tunnel that something could happen if three or four teams drop points.”
Liverpool are in seventh place in the Premier League table after their 2-1 victory over Palace.
The Reds welcome Manchester United in the first leg of their Europa League last-16 tie at Anfield on Thursday night.
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