The Reds came from behind after a 57th-minite strike from Jean Philippe-Mateta whose low-drivem shot sent Alisson the wrong way.
The home team was earlier denied a penalty appeal in the first half after a controversial Video Assistant Referee check.
In the 75th minute, Ghanaian attacker, Jordan Ayew, pulled down Liverpool’s Elliot at the centre of the park, before getting sent off after his previous booking.
A minute later, Mohamed Salah struck a deflected shot after receiving a pass from Curtis Jones.
Salah marks a milestone as he registered his 200th goal for the club.
Right at extra time, Elliot struck the ball into the bottom-right corner of Palace’s net as he received a 30-yard pass from Salah.
Palace manager, Roy Hodgson made two changes at the start of the second half after a 0-0 game in the first 45 minutes.
Since their 2-0 win at Burnley, Palace has won just once in their last eight league games. They occupy the 14th position with 16 points from their 16 matches played.
Liverpool have remained unbeaten in their last eight matches and overtake Arsenal to stay atop the league table.
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