Jurgen Klopp's men would have needed favours from Leicester and Chelsea in their final two games of the campaign had they failed to beat the already relegated Baggies.
But the Brazilian was sent forward for a corner deep into stoppage time and headed home Trent Alexander-Arnold's delivery.
Hal Robson Kanu had put the Baggies ahead with 15 minutes on the clock before Mo Salah levelled on 33 minutes.
Alisson's late goal meant Liverpool moved one point behind fourth-placed Chelsea and three behind Leicester.
With Leicester and Chelsea still to face off in a repeat of the FA Cup final, won 1-0 by the Foxes on Saturday, victories over Burnley and Crystal Palace will take Liverpool into the top four.