Cup kings Burnley dump Baggies out of FA Cup
The Championship side had already beaten Arsenal, Chelsea and Fulham in the League Cup before falling to Tottenham Hotspur in the semi-finals.
Midfielder Wade Elliott put Burnley ahead just before halftime at Turf Moor, striking the ball from just outside the box into the top left corner, before Thompson stretched their lead seven minutes after the break.
The Premier League's bottom side, who shared a 2-2 draw with Burnley at the first attempt, got themselves back into the game in the 60th minute when Dutch defender Gianni Zuiverloon shot from the edge of the penalty area into the bottom left.
Thompson put the game out of West Brom's reach with his second into the bottom left two minutes from time.
Burnley will go on to face Arsenal or Championship side Cardiff City, whose replay was postponed due to bad weather.
There is more FA Cup Fourth-Round replay action on Wednesday when Everton host Merseyside rivals Liverpool and Sunderland visit Blackburn Rovers.
Championship side Doncaster Rovers will seek an upset at Aston Villa, while Nottingham Forest and Derby County meet again in an all-Championship replay.