But the visitors were not about to roll over and they fought back to level at 2-2 in the 65th when James Tomkins was left unattended to head in a Luka Milivojevic corner.
In the past, Liverpool have been criticised for falling at the final hurdle and actually, they did so against Palace back in 2014.
James Tomkins headed Palace level but an bad error by veteran goalkeeper Speroni, tipping James Milner's cross towards his own goal for Salah to score his 50th goal in the Premier League, gave Liverpool a lead that was later added to by Sadio Mane.
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"When he announced he was retiring I was like "wow, you could have another three or four years the way that you're playing". A wonderful goalkeeper, professional and gentleman who has been an absolute credit to our game.
"We needed to improve a few things and do things again and again".
He is the fourth-fastest player to reach the 50-goal landmark in the Premier League, behind only Andy Cole, Alan Shearer and Ruud van Nistelrooy.
"Wilfried Zaha is a world-class player and he gets into a one-on-one situation and we're one down".
Palace still had time to get a third with Meyer placing a shot past Alisson from 15 yards out after good work from Connor Wickham, but Liverpool were able to hold on to register what could prove to be a significant result come the end of the campaign.
Liverpool's back line then handled some late pressure from Palace to seal the three points.
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Former Liverpool manager Roy Hodgson had set Palace up perfectly to do a similar job to the one the south London side delivered in their victory at Manchester City last month. "There are different ways to win games and today we had to dig in with all we had".
Liverpool trailed 1-0 at half-time against Palace before fighting back in the second half. "So I'm not sure what's insane about it, I'm disappointed for the boys that they've got got anything from it", Hodgson said.
Salah himself insisted Liverpool deserved to take all three points, although he conceded allowing Palace to score three goals was far from ideal.
"You need character and personality and good team spirit, people prepared to stick to the game plan even when it's not going your way".
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