The attacker levelled for Liverpool not long after the break with a neat finish from James Milner's low cross, before Mohamed Salah showed great composure to lash home the victor with just six minutes left on the clock.
The league's leading scorer kept his cool to convert Andrew Robertson's cross on 84 minutes for his 29th goal of the season.
The 84th-minute strike, which followed Sadio Mane's equaliser, turned the table around and left Palace two points above the relegation zone.
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However, they bounced back in convincing fashion, as Mohamed Salah netted in a cool finish, in the dying stages of the second half.
Next up for Liverpool is City at home in the Champions League and Klopp thinks they are going into the game in a great mood as a result of the comeback.
Andy Robertson absolutely loved Mohamed Salah's late victor for Liverpool at Crystal Palace.
But Liverpool's first-half performance did not match the boldness of Klopp's selection as they lacked intensity and imagination against a stubborn Palace defence.
As in their defeat to United at Old Trafford earlier this month, Liverpool were exposed down their right as young full-back Trent Alexander-Arnold struggled to cope with the pace of power of Wilfried Zaha.
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An injury to Adam Lallana shortly after coming forced a change to three at the back which worked well to force our full backs forward which led to Mo Salah's late victor.
The former Liverpool striker sent an attempted lob well wide with just Karius to beat and then shot over the bar after Van Dijk had been caught in possession by Andros Townsend.
Salah's also about to steal a record that has been the domain of Manchester United players so far, something that will no doubt make Liverpool fans happy.
Wayne Hennessey - 6 - Did his job for most of the game and wasn't at fault for the goals.
Corey Feldman attacked, stabbed in the stomach
But of Feldman's injuries, Madison says that there "is no laceration or stabbing to his stomach". While security was distracted... a guy pulled up and attacked.
At the other end, Sadio Mane was unfortunate to be awarded a yellow card for simulation.