Palace took a 13th-minute lead through Luka Milivojevic, who stroked home a penalty after Wilfried Zaha was taken out by Liverpool goalkeeper Loris Karius.
After a frustrating first period, Sadio Mane got the Reds back on level terms four minutes after the restart, as he turned in James Milner's cross from the left flank. The forward then somehow avoided a second yellow card following a blatant deliberate handball.
It was a moment the home side were left to rue, as with six minutes remaining, Salah controlled an Andrew Robertson cross expertly and slotted home his 37th goal of a remarkable campaign.
Zaha Leaves Alexander-Arnold in a Daze
The early stages of the match were defined by the battle down Palace's left flank, as Zaha went head-to-head with Trent Alexander-Arnold. There was only one winner in this early tussle.
Zaha constantly escaped the attentions of the youngster and should have put the Eagles ahead prior to Milivojevic's penalty, only to be denied by Karius.
As noted by the Guardian's Sachin Nakrani, it was a mismatch on the wing:
The penalty came after Zaha latched on to a header from Christian Benteke and was poleaxed by the onrushing Liverpool goalkeeper; the referee had no option but to point to the spot, and Milivojevic stroked it home.
Mane Scores, Somehow Escapes Red
The man most involved in an attacking sense for Liverpool was Mane, although things didn't go well for the Senegalese in the opening 45 minutes.
First, he should have done better with a flick header from close range after Virgil van Dijk's knockdown. Then Mane took a dive in the penalty area after a small touch from James McArthur.
As Seb Stafford-Bloor noted, the plunge from the Liverpool player was pretty comical:
At the end of the first period, Mane almost broght his side level as he diverted a powerful header towards goal. However, Wayne Hennessey got down well to keep out the effort and keep Palace's nose in front at the break.
The forward was not to be denied after half-time, though, as he got in front of Mamadou Sakho to tuck home from close range and reinvigorate the away side.
Melissa Reddy praised Mane's persistence:
After putting Liverpool level, the forward was then fortunate to stay on the field, as he knocked the ball away from a Palace player with his hand. To the incredulity of the Palace players, manager and fans, a second yellow wasn't awarded. Understandably, Reds boss Jurgen Klopp withdrew the player not long after.
Salah Shows Benteke How It's Done
It was Palace who had the best chances to retake the lead following the equaliser, although unfortunately for the Eagles, two clear openings fell to the hapless Benteke.
Twice he was through on goal with only Karius to beat, but he skewed the first chance wildly over the bar and tamely struck the second at the Liverpool goalkeeper.
Colin Millar didn't mince his words in assessing the Belgian's display:
At the other end, the Premier League's top goalscorer only needed one opportunity to win the game for Liverpool.
When the ball dropped at Salah's feet in the box, he took a brilliant touch to knock Sakho off balance before calmly placing the ball past Hennessey.
Next up for Klopp's side is a showdown with Manchester City in the UEFA Champions League quarter-final first leg at Anfield on Wednesday. As for Palace, their quest to avoid the drop will continue away at Bournemouth on Saturday.