The Reds needed a win to keep their hopes of a top four finish alive.
But it looked as if they would come up short with the game 1-1 going into the fifth minute of stoppage-time.
Liverpool were given a corner and Alisson Becker was given the go ahead to get in the box.
And, unbelievably, he managed to find the net with a superb header, sparking jubilant scenes.
The result could have been very different, though.
West Brom thought they had taken the lead with 20 minutes remaining when Kyle Bartley diverted the ball home.
However, the goal was controversially ruled out as Matt Phillips was standing in an offside position.