The striker climbed off the bench to score twice in extra time and clinch a 3-1 fourth-round replay win.
The Seagulls will host Derby next week after coming from behind to win at The Hawthorns.
Kyle Bartley's bizarre goal, when the defender hooked in from almost behind the goal, opened the scoring late on.
But Brighton's Florin Andone levelled with eight minutes left having escaped a first-half red card following an elbow on Sam Field.
Leon Balogun and West Brom's Hal Robson-Kanu hit the bar in the first half for either side in front of just 8,645 at The Hawthorns.
Brighton deserved the victory but needed David Button to save Jay Rodriguez's angled half-volley after eight minutes.
Six minutes later the striker had Button scrambling to turn his low 25-yard effort wide.
Yves Bissouma followed suit when a similar strike forced Jonathan Bond into action.
But the game lacked clear chances and a subdued Hawthorns only came to life when Andone escaped a red card after catching Field in the 31st minute.
The pair tussled for a bouncing ball in the centre circle before Andone planted an elbow into Field's face.
It was missed by referee Paul Tierney and Andone twice went close to opening the scoring soon after.
First, Bond saved low from Viktor Gyokeres and Andone's deflected follow-up flew over. From the resulting corner Balogun crashed a header against the bar.
Brighton kept pressing and Andone dragged inches wide after being sent through by Anthony Knockaert eight minutes before the break.
The Seagulls were on top and Ali Jahanbakhsh's curling effort tested Bond after the restart.
West Brom had withdrawn Rodriguez for defender Craig Dawson at half-time and struggled to impose themselves as Brighton remained on top.
The Seagulls should have had a penalty with 25 minutes left when Tyrone Mears brought down Jahanbakhsh but their justified appeals were ignored.
Dan Burn headed wide but against the run of play the Baggies opened the scoring with 13 minutes left.
It was a strange goal as Bartley tapped in while most of his body was behind the post after Balogun thought Hoolahan's free-kick, which Burn had flicked on, was dropping wide.
Suddenly the hosts were pressing and Rayhaan Tulloch should have made it 2-0 but planted a free header wide from six yards.
The Baggies were made to pay as Andone levelled with eight minutes of normal time left when he drilled under Bond following Dawson's slip.
It meant extra time and Albion finished with 10 men after Robson-Kanu limped off with the Baggies having made all their substitutes before Murray proved the difference.
The forward, who had only been on the pitch three minutes, struck in the 104th minute as he bundled in from close range when Ahmed Hegazi failed to cut out Locadia's cross.
He then sealed victory after 117 minutes, lifting past Bond from close range.
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