Mohamed Salah's 37th goal of the season in all competitions handed Jurgen Klopp victory in his 100th Premier League match in charge of Liverpool.
Palace, who inflicted the first league defeat of Klopp's tenure, were left frustrated in their battle to stay alive in the Premier League, seeing Liverpool come from behind and Sadio Mane escape a red card in the second half.
Luka Milivojevic gave Palace the lead with a 12th-minute penalty before Mane equalised just after half time. But Mane was fortunate to stay on the pitch after he deliberately handled the ball while already on a booking and Salah then struck six minutes from time to seal an important win at both ends of the table.
Palace remain mired in a relegation fight while Liverpool moved 10 points clear of Chelsea in fifth, with Champions League qualification for next season firmly within their grasp.
Klopp's men embarked on a crucial 10-day spell in their season at Selhurst Park, with a two-legged Champions League quarterfinal against Manchester City looming as well as a trip to Everton for the Merseyside derby.
But the Liverpool coach resisted the temptation to rest his key men in South London, with Salah, Mane and Roberto Firmino all starting in attack.
And despite the visitors dominating possession in the early stages, it was Palace who fashioned the game's opening chance with Wilfried Zaha finding space in the area, only to be brilliantly denied by Loris Karius.
Zaha, an injury doubt before the game, met a long ball with Trent Alexander-Arnold asleep at the back but Karius bailed his teammate out with a commanding stop. Soon after, Virgil van Dijk's free header from a corner appeared to be heading in, only for Mane to flick the ball wide.
Liverpool's defensive deficiencies reared its head for the opener, when they failed to deal with another long ball forward, nodded on by Christian Benteke to the advancing Zaha. Karius clattered him in the area and Neil Swarbrick had no hesitation in pointing to the spot. Milivojevic stepped up to send Karius the wrong way for his seventh converted penalty out of eight this season -- more than any other player.
Liverpool then wanted a penalty of their own when Mane went down in the area just past the midway point but after consulting with his assistant, ref Swarbrick punished the Liverpool forward with a yellow card for simulation.
The away side's frustration grew when Mane's header was ruled out for offside but he didn't have to wait long to finally get on the scoresheet. Just four minutes after the restart, Mane met James Milner's driven ball in from the left to tap home the equaliser.
Benteke, with seven goals in nine appearances against former club Liverpool, then squandered two glorious opportunities in the space of a minute to put Palace back in front. First, he made a real mess of a lob over Karius inside the area before blasting over from a similar position.
With Palace in the ascendancy, Liverpool escaped on the hour when Mane, who thought he was fouled by Andros Townsend, then grabbed the ball with his hands. Swarbrick awarded a free kick in Palace's favour but opted against handing Mane a second yellow card for deliberate hand ball.
Liverpool then made Palace pay in the closing minutes, Salah sliding home the winner from close range to seal the win.