Liverpool also suffered another injury concern when Joel Matip limped off with a groin injury in the second half.
West Brom earned their first point under new boss Sam Allardyce but remain in the relegation zone.
Liverpool dominated possession from the first minute and opened the scoring after 12 minutes when Matip clipped the ball into Mane who chested it down and rifled a shot past West Brom goalkeeper Sam Johnstone.
By half-time, Liverpool had enjoyed 87% possession and no West Brom player had completed more than eight passes.
The visitors provided a better threat in the second half and were given a boost after Matip limped off after 59 minutes.
Liverpool continued to be frustrated and West Brom almost grabbed an equaliser when Karlan Grant was put through but Alisson blocked his shot
West Brom grabbed their equaliser after 82 minutes when Ajayi headed in from Matheus Pereira‘s cross.
Roberto Firmino had a wonderful chance to win the game in stoppage time but saw his header saved by Johnstone.