Steve Thompson was the hero for the home side with two goals as Owen Coyle's men claimed another memorable top-flight scalp.
Wade Elliott volleyed them ahead on the stroke of half-time before Thompson fired home from close-range.
Gianni Zuiverloon's goal gave West Brom some hope but Burnley made sure of their place in the next round as Thompson grabbed his second.
Coyle's impressive Coca-Cola Championship outfit bundled Fulham, Chelsea and Arsenal out of the Carling Cup and came agonisingly close to overcoming Tottenham in an epic semi-final encounter.
With one Wembley dream having died, another remains firmly alive after this latest success which left Albion to focus solely on their grim fight to avoid relegation.
This was the 41st game of Burnley's marathon season but they were never going to be overawed by a Baggies side who stand just eight league places above them.
Bright startThe Clarets fans who braved the freezing temperatures were warmed by the prospect of a possible trip to Emirates Stadium in the next round as Burnley will face Arsenal or Cardiff.
In the 12th minute, Robbie Blake's low strike from distance was palmed around a post for a corner by Scott Carson before Roman Bednar cleverly engineered the space to drive a cross-shot hopelessly off target.
The Baggies continued to produce passages of play in keeping with their reputation as football purists, but Burnley's defence held firm.
Blake's trickery with the ball emerged as Burnley's most potent attacking threat and he twice crafted the space to curl right-footed shots just wide of Carson's goal.
Albion, so typically, kept the ball alive at every opportunity around their opponents' penalty area without ever really threatening to score and Burnley were not much more menacing at the other end.
The visitors were dealt a blow in the 36th minute when on-loan Arsenal striker Jay Simpson, who had looked lively, picked up a knock and was replaced by Chris Brunt.
However, Albion carved Burnley open four minutes before the interval when Marc-Antoine Fortune beat the offside trap and laid the ball back to Bednar, whose vicious, rising strike from 14 yards was diverted for a corner by Brian Jensen.
Albion defender Ryan Donk then saw his far-post header smartly saved by Jensen before Burnley went ahead.
OpenerA long punt upfield from Jensen was headed up into the air by Donk and Elliott was on hand to strike a superb volley into the far left-hand corner from 20 yards out.
Bednar headed weakly wide when well placed within two minutes of the restart and Burnley soon capitalised by doubling their advantage in the 52nd minute.
Christian Kalvenes crossed dangerously from the left flank and although Martin Paterson's far-post header was blocked by a Baggies defender, Thompson lashed in the loose ball right-footed from six yards out.
Yet the visitors responded impressively and when Zuiverloon was given space on the right side of the Burnley penalty area, he expertly drilled a fine low shot past Jensen and into the bottom corner.
A third goal should have arrived for Burnley in the 70th minute when Blake's long-range free-kick was parried weakly by Carson.
The ball fell straight into the path of the lurking Paterson but he was not sharp enough and the chance of a simple tap-in was spurned.
Albion pushed hard for an equaliser but their few hundred fans in a crowd of just 6,635 were left disappointed as Burnley added a third when Blake's brilliance saw him cross low from the right to give Thompson the easiest of tap-ins.