An early goal from Dan Agyei was enough to take the Clarets into the last 16 of the Premier League Cup as they completed their great escape from the group stage.
An opening 7-1 defeat at Bournemouth made it look a long way to the knock-out rounds.
But a brilliant unbeaten run since saw Steve Stone’s side top the group to set up a home tie next.
Going into their final game, the Clarets were sitting in bottom spot and knew nothing less than victory would do them.
That was the same scenario facing West Brom as Liverpool waited in the wings to sneak through with Bournemouth in the event of a draw.
And with just one goal in it from the second minute onwards that was always a possibility.
But with Agyei’s 21st goal of the season as a cushion, the Clarets produced a magnificently determined and disciplined performance, punctuated with further chances, to hold their nerve and book their passage through.
The Clarets couldn’t have asked for a better start than that given to them by Agyei inside the opening two minutes.
The leading scorer has been in a rich vein of form in recent games and continued it when he coolly tucked away a beautifully whipped-in low cross from Tunde Bayode.
And it could have been 2-0 moments later when Agyei again tore a hole in the Baggies’ defence with a storming run only stopped by a fine save from Alex Palmer.
The Albion keeper was to make another crucial save to deny Agyei again before the opening half was out after Rob Harker had laid off another telling delivery from Bayode.
A second goal would have given Burnley valuable breathing space and left the visitors with a mountain to climb to qualify.
But with little margin for error it needed a sharp save from Adam Legzdins to keep the Clarets in the driving seat when he kept out a fierce strike from dangerman Jonathan Leko.
The England Under-19 international – with a total of 27 first-team appearances to his name from his time at the Hawthorns and on loan at Bristol City – again carried the fight for the Baggies at the start of the second half.
He whistled a long-range effort just wide, but the Clarets refused to buckle with Legzdins called into serious action just once more to make an alert stop to keep out an angled effort from Rayhaan Tulloch as he ensdured a second successive clean sheet in the competition.
Burnley looked to relieve the pressure with 16-year-old substitute Lewis Richardson involved in their best move of the half which ended with Agyei squeezing in a shot which is parried by Palmer.
Agyei was always willing outlet as timed ticked away and West Brom poured forward in search of the goal that would given them a glimmer of hopes and dashed the Clarets’ dreams.
But they were never allowed a way through as a defence coaxed and cajoled by Legzdins – behind a midfield which admirably made up for the absence of the injured Josh Benson – performed superbly to make it six games unbeaten in league and cup and keep Stone’s young charges competing on two fronts.
Burnley: Legzdins; Cropper, Thomas, Wilson, Glennon; Bayode, Wilson, N’Guessan (Perkins 77), Blair (Chakwana HT); Agyei, Harker (Richardson 66). Substitutes: Stone, Clarke.
Booked: N’Guessan, Glennon
West Bromwich Albion: Palmer; Ferguson, Solanke, Healy; Gardner-Hickman, Rogers, Azaz, Melbourne; Leko (Asomugha 86), Tulloch, Edwards. Substitutes: Brown, House, Gilbert, Delaney.
Booked: Tulloch, Ferguson, Rogers
Referee: Garreth Rhodes