Jurgen Klopp's side remain in fifth place but are now a point behind Chelsea with two games to go.
West Brom took the lead after 15 minutes when Robson-Kanu produced a fine curling finish past Alisson, scoring in his first Premier League start since 2017.
Mane should have equalised less than 10 minutes later but his close-range effort went past the post.
Liverpool did level the score after 32 minutes after Salah's strike from outside of the area found the bottom corner.
The visitors came close to a second moments later, but Roberto Firmino's shot rattled the crossbar.
Mane thought he had put Liverpool ahead four minutes after the break, but his goal was ruled out for offside.
Just before the hour mark, Salah found himself on the end of a swift counterattack, but his shot was saved by Sam Johnstone.
But in the 94th minute, Alisson came up and grabbed a stunning winner from a corner to keep Liverpool in the race for the top four.