Wales midfielder Joe Ledley handed Palace a second-half lead against the run of play after Liverpool had dominated the opening 45 minutes at Selhurst Park.
Liverpool vice-captain James Milner was sent off shortly after Ledley’s opener to leave Klopp’s side facing a fourth defeat in five games against Palace.
But Roberto Firmino capitalised on Alex McCarthy’s slip to level before substitute Christian Benteke converted an added-time penalty to seal a 2-1 victory.
Liverpool’s triumph in the English capital was the first time the Reds have secured three successive Premier League victories this season.
Champions League winner Carragher explained why he felt Jurgen Klopp’s side would more than likely miss out on a top-four finish this term.
“There is a slight possibility. There are big outsiders to get that spot,” Sky Sports.
“You look at the form of West Ham and Man City. You expect them to win at home but can you trust them away from home?
“Liverpool have been very inconsistent and this is the first time they’ve won three on the bounce. They’d have to win six or seven games in their last 10 games to do that.”
Liverpool are in seventh place and six points behind fourth-placed Manchester City, who occupy the final Champions League qualifying spot.
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