With two-thirds of the league season in the books, West Brom is in serious danger of relegation-the side sits 19th with 20 points, three points from safety.
Matip's own-goal was confirmed, to make it 3-1, following a delay of nearly two minutes, with the question having been over whether there had been an offside. Salah pulled a goal back in the 78th minute with a well-hit shot into the bottom corner.
Jonny Evans was perhaps a surprise starter given the speculation concerning Arsenal, and the uncertainty over his future seemed to impact his decision-making on the pitch early on.
Barry did make an attempt to play the ball, and was interfering with play in front of goalkeeper Simon Mignolet, and the correct decision was given just over a minute after the original decision.
With just over 10 minutes played, Rodriguez had his second.
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The Baggies soon had the ball in the net again through Craig Dawson, but the defender's celebrations were halted by referee (and almost-namesake) Craig Pawson, who chatted to colleague Andre Mariner before ruling it out.
In between, Liverpool were awarded a penalty after a prolonged VAR review, only for Roberto Firmino's spot-kick to come back off the crossbar. Four minutes were added on as stoppage time in total in the first half.
"We had to show character in the second-half".
Foster saved well from James Milner and Salah gave Liverpool some late hope with a cool finish from 12 yards after West Brom failed to clear, but the visitors held firm for a win that continues their improved form under Alan Pardew.