The Reds found themselves behind when Palace struck on the counter-attack, with the ball crossed to Andros Townsend just inside the area for the winger to lash home.
Liverpool boss Jurgen Klopp had some work to do at the break but it did not take long before his side were back on level terms when Virgil van Dijk struck a speculative effort from range, only for the deflection to loop into the path of Salah who nudged the ball home.
The hosts found themselves in the lead for the first time on 53 minutes when Roberto Firmino worked himself some space in the area and placed the ball into the bottom corner.
The Anfield crowd were stunned on 65 minutes when James Tomkins was given a free header in the box from a simple corner and the Eagles were back on level terms.
However, it was not long before Liverpool’s lead was restored when Salah popped up with his second of the game when James Milner crossed the ball before the Egyptian tapped home from a yard out.
It was a nervy end when Milner was given his marching orders for a second yellow although the Reds went on to score again thanks to a late effort from Sadio Mane.
Palace pulled another goal back, with Max Mayer getting on the scoresheet, but it was not enough to stop Liverpool extending their lead at the top of the Premier League back to seven points.