Liverpool took the lead after 20 minutes through Daniel Sturridge, who was making his first Premier League start since October.
But it only lasted five minutes. Didier N’Dong went down in the box after a collision with Liverpool defender Ragnar Klavan.
And Jermain Defoe converted his penalty cooly past Reds goalkeeper Simon Mignolet, who failed to stop the equaliser despite going the right way.
The visitors were back in front on 72 minutes after a corner flicked off N’Dong’s head into the path of Sadio Mane to hit home from six yards.
But Defoe again scored from the penalty spot with five minutes left on the clock to clinch a valuable point in Sunderland’s relegation scrap.
Liverpool goalscorer Mane handled the ball in the box for referee Anthony Taylor to point to the spot.