The 26-year-old continued from where he stopped last week, helping Jurgen Klopp's men overcome the stern test of Wilfried Zaha and his counterpart.
Zaha, a constant nuisance to the Reds down the right, orchestrated an opener for Palace against the run of play in the 34th minute.
The Cote d'Ivoire international whisked past James Milner, squaring for Andros Townsend to stun the hosts.
Salah started the fightback for the Reds immediately after the restart in an improvised manner.
Virgil van Dijk's long-range shot was deflected, looping behind Palace’s defensive line and before Julian Speroni could decide on what to do, Salah provided a poacher's touch to put his side levelled.
In the 53rd minute, Naby Keita provided the assist for Roberto Firmino's goal which put the Reds in front for the first time in the game. But they were dragged back again shortly after the hour mark as James Tomkins powered home from a corner.
Salah would, however, score the decisive goal in the game, crashing home from a Speroni error, via Milner's cross.
In the stoppage time, Max Meyer cancelled out Sadio Mane late effort to set up a nervy ending to the thrilling encounter that Liverpool finished with 10 men after Milner was shown a red card for catching Zaha late.
The brace helped Salah to 50 Premier League goals and were crucial for the Kops to extend their home unbeaten run to 31 games.
Pending Manchester City’s game on Sunday, they are seven points clear at the top of the table.