Baggies boss Tony Mowbray was left to rue a wasteful night in front of goal as Albion added their name to Burnley's giant-killing list this season.
Although controlling the majority of the game with former stopper, Brian Jensen, making smart first-half saves from Roman Bednar and Ryan Donk, Albion found themselves behind after the break after a fine volley from Wade Elliott.
The second-half didn't start much better for the Baggies as Steven Thompson extended the home side's advantage seven minutes after the break. Dutch defender Gianni Zuiverloon scored his first goal in an Albion shirt to give the Baggies a lifeline, but as Mowbray's men pushed forward for an equaliser, a twisting run from Robbie Blake set up Thompson in the 88th minute for his second of the night.
"I have to give Burnley credit," the Baggies boss said.
"They worked extremely hard, got the goals and are through to the next round.
"We wish them luck.
"We missed a few opportunities and we were generally on top in the first half but found ourselves behind to a great strike.
"I was sat there feeling pretty comfortable because I thought we were the better side.
"But it doesn't matter because football is all about putting the ball into the net.
"I think Burnley's strikers showed great desire and quality in the final third, which was probably the difference.
"We lacked that clinical touch which has dogged us all season.
"We are not as clinical as we should be in the final third.
"We have to take it on the chin now and move on.
"The league becomes the priority, as it always has been.
"You want to go as far as you can in the cup but that distraction has gone now.
"I'm disappointed to be out of the cup but not overly disappointed.
"It's just a bit frustrating to go out to a team I thought we could dominate.
"Generally we did - yet we lost three goals."
With Albion's mounting injury list Mowbray made six changes to his starting XI; however still saw Jay Simpson limp off in the first-half with a hamstring problem. The on loan forward was immediately substituted with Mowbray optimistic the pacy 20-year-old will be fit for Saturday's fixture against Newcastle (KO 3pm).
"It looks like Jay has a slight strain of his hamstring from overstretching, which doesn't help the cause," Mowbray continued.
"We didn't want to take a chance and we got him off as quickly as we could.
"I hope he'll still have a chance for the weekend because we got him off so quickly.
"You never know with hamstrings."