The Merseysiders' hopes of a place in next season's Champions League look over as the Baggies soaked up the pressure before striking twice late on to secure their first win in 2013
Gareth McAuley and Romelu Lukaku put West Brom on the board after Steven Gerrard missed a penalty.
Brendan Rodgers was forced into making one change from the XI that took a point away from Manchester City, with Jonjo Shelvey coming in for the injured Daniel Sturridge while for West Brom, Liam Ridgewell and Yousouff Mulumbu returned to stiffen up the visitors.
The early stages were far from frenetic but as usual it was Suarez who was the first to threaten with a flashed volley that drifted wide of Ben Foster’s right-hand post before Shelvey stroked home a rebound only for the effort to be ruled out for offside.
The hosts were in control and could have taken the lead when first a free header from Daniel Agger was tipped over the bar well by Foster before Stewart Downing, an impressive performer throughout, saw a fierce left-footed effort from long-range sting the hands of Foster.
Opportunities for the out-of-form Baggies to break were few and far between with Agger next to threaten on the half-hour mark, failing to make contact with his head despite having the freedom of the West Brom box.
Steven Reid then came close to an embarrassing own goal when he horribly sliced a drilled Shelvey cross that only just cleared the top of his crossbar as the two sides closed out a poor first half on level terms.
The second half began in much the same vein as the first ended with Foster making a sprawling blocked save to deny first Downing and then Gerrard from breaking the deadlock. Minutes later, Jordan Henderson’s improvised back-flick finish brought good reactions from the excellent former England goalkeeper.
On the hour mark Rodgers reacted by bringing on Fabio Borini and youngster Raheem Sterling as he looked to find more of a cutting edge and with just over 20 minutes on the clock, Borini went close to scoring his first Premier League goal for the Merseysiders but again found Foster equal to the effort.
The Baggies again had Foster to thank when the keeper made a super one-handed save from a drilled Gerrard effort, with the follow-up put wide of the post by the onrushing Borini. The best was still to come from the former Manchester United shotstopper when Liverpool was awarded a penalty after Olsson was adjudged to have felled Suarez with 15 minutes to go. Gerrard stepped up but saw his spot kick superbly stopped by a diving Foster.
That miss would come back to haunt the home side when first Mulumbu’s daisy-cutting long-range strike was tipped wide by Pepe Reina before from the resulting corner, McAuley thumped in a header off the bar.
Liverpool was stunned and its misery was complete in stoppage time when substitute Lukaku ran through and slotted past Reina.