Divock Origi's deflected 95th-minute strike rescued a point for Liverpool against a resolute West Brom.
The home side began well and took the lead when Jordan Henderson finished a superb and patient passing move.
But Simon Mignolet's error gifted a goal to Craig Dawson of Albion, who were later furious when a delayed offside flag denied Olsson a goal.
Olsson looked to have headed West Brom to victory before substitute Origi's shot found the net deep in added time.
Such was the relief around Anfield at full-time that Reds boss Jurgen Klopp called his entire team together to conduct on-field celebrations in front of the Kop.
The point earned means Liverpool are ninth in the table, five points off the top four, while the two points lost leave West Brom 13th, six points above the bottom three.