It marked the second time in the space of four days that Jurgen Klopp’s side had to come from behind to take the points, after their midweek win over Manchester United.
The Reds needed to keep winning to keep the pressure on Chelsea and Leicester, but once again struggled to find a clinical edge against a team at the bottom, despite Mohamed Salah keeping pace with Harry Kane in the race for the Golden Boot.
An early shot from Thiago Alcantara - the game’s stand-out performer - and a flicked header from Curtis Jones signalled the Reds’ intent to attack, though the Baggies had their own half-sights of goal via set-pieces.
It was open play, though, and a moment of disorganisation in the makeshift defence of the away side which led to the opener; Fabinho lost an aerial battle, Rhys Williams stepped out to challenge, Nat Phillips was deeper to play Hal Robson-Kanu onside.
The Welshman’s finish was exquisite, curled into the far bottom corner beyond the reach of Alisson Becker.
Sadio Mane was presented with a great opportunity to equalise after a flowing move with Thiago at its heart, but he drilled his shot wide of the near post. The No. 6 then fizzed a pass in to allow Trent Alexander-Arnold a shooting chance - this one over the bar - and an equaliser wasn’t much longer in coming.
The recalled Mane it was who made the goal, playing forward quickly after a free-kick and then winning a tackle on the edge of the West Brom box - but the strike from Salah was unerring, first tie into the far bottom corner.
Within minutes it might have been 2-1 to either side; Roberto Firmino smashed a shot on the turn onto the post, before another defensive mix-up between Phillips and Williams let Robson-Kanu through again, Alexander-Arnold this time coming to the rescue with a last-ditch tackle.
Mane slid in another just after the restart, only to be flagged offside, before Salah spurned a one-on-one after a lightning break following a West Brom corner.
West Brom remained a threat off set-pieces and on the counter, though, as the Reds pushed forward for the second goal they needed to keep their Champions League hopes alive - and they were indebted to an officiating call too, after Kyle Bartley’s prodded finish was ruled out due to Matt Phillips obstructing Alisson’s view from an offside position.
Jurgen Klopp threw on Xherdan Shaqiri and Gini Wijnaldum in the search for a late winner and the latter had the best chance - but fired high and wide from the edge of the box in the closing minutes.
Just as it seemed all hope was lost and that Liverpool would be all-but-out of the race for the top four once more, Alisson went up for a 94th-minute corner - and rose highest to plant a brilliant effort into the far corner and save the day for his team.
The Reds are now one point off Chelsea and three behind Leicester, with the FA Cup finalists due to face each other on Tuesday night.