Brendan Rodgers' side led by 3-0 with 11 minutes to go, but two goals from substitute Dwight Gayle and one from Damien Delaney gave Palace a share of the points.
The result means Liverpool are one point clear of Manchester City, with City having two games left to Liverpool's one.
Goals from Joe Allen, Daniel Sturridge and Luis Suarez had Liverpool in cruise control, but Palace manager Tony Pulis brought on Dwight Gayle, and it changed the game.
Delaney pulled one back with a deflected strike, and Gayle scored a second on the counterattack, and Liverpool's defence crumbled as Gayle slotted in a third.
City can now go two points clear if they beat Aston Villa on Wednesday.