The victory is Burnley's fourth against top flight opposition this season and goes some small way towards exorcising the ghost of their last-gasp defeat to Tottenham in the Carling Cup semi-final.
West Brom boss Tony Mowbray made six changes to the side that faced Hull in the Premier League last Saturday, indicating where the FA Cup features in his list of priorities.
It was one of those changes, Roman Bednar, who almost gave the visitors the lead four minutes before the interval when Marc-Antoine Fortune beat the offside trap and laid the ball back to the Czech, but his strike was diverted for a corner by keeper Brian Jensen.
Three minutes later, Burnley took the lead when a long punt upfield from Jensen was headed upwards by an Albion defender and Elliott was on hand to strike a superb volley into the left-hand corner from 20 yards.
Burnley grabbed a second seven minutes after the break when Martin Paterson's far-post header from a Christian Kalvenes cross was blocked, but Thompson was on hand to lash the loose ball in from six yards.
Yet the visitors responded impressively and with half an hour to go Zuiverloon was given enough space on the right side of the Burnley penalty area to drill a fine low shot past Jensen into the bottom corner.
Any hopes the Baggies had of forcing extra-time were ended two minutes from time when Blake brilliantly set up Thompson in the six-yard box to tap in his second and seal a deserved victory.
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