Palace's first chance was squandered by Jordon Mutch with 20 minutes gone, with his effort hitting the side-netting after being set up by Connor Wickham.
Two minutes later the midfielder had a penalty appeal turned down after going down in the box.
But the setback did not hinder Sam Allardyce's side and they took the lead when Dame N'Doye lashed home with the help of a massive deflection off Scott Dann to score his first goal for the club in the 36th minute.
Jermain Defoe had a couple of chances to increase their lead but the home side had to settle being one goal to the good at the break.
Wickham had an early opportunity at the start of the second half but his header was directed over the bar.
The former Sunderland man equalised for Palace in the 60th minute when he produced a neat turn and swivel before seeing his shot cannon off the post to make the score 1-1.
Wickham struck again just five minutes later to put his team ahead with a thumping volley from inside the six-yard box making it two braces in two matches after the weekend's double at West Bromwich Albion.
Sunderland introduced Borini with 10 minutes left, and the former Liverpool forward drew the Wearsiders level in the 90th minute with a quite brilliant strike from the right side of the area across Hennessey.
The result has Sunderland out of the drop zone, albeit momentarily as local rivals Newcastle United have two games in hand, while Alan Pardew's Palace remained 14th following an 11th league match without a win.
2015/16 Barclays Premier League - season scores