Martin Patterson makes it 2-2 with just two minutes to spare
Burnley came from behind late-on to force a replay and go into the hat for the FA Cup fifth-round draw.
Graham Alexander thumped Burnley into the lead from the penalty spot after Jonathan Greening had pushed Robbie Blake over in the area.
Robert Koren pulled West Brom level, crashing home a 25-yard volley before Do-heon Kim's deflected free-kick put West Brom ahead in the first-half.
Burnley levelled on 89 minutes when Martin Paterson turned in a low cross.
A buoyant Burnley showed no signs of fatigue after their mid-week Carling Cup semi-final game against Tottenham and stunned the Premier League side by taking the lead early on at The Hawthorns.
The Clarets have already beaten Premier League sides Fulham, Chelsea and Arsenal in the League Cup this season but had to come from behind late on to keep their FA Cup hopes alive.
The opening goal came when West Brom captain Greening was adjudged to have fouled Blake and Burnley's spot-kick specialist Alexander stepped up to drill home a rising shot which Scott Carson had no chance with.
Greening stroked a corner into the Burnley box which Steven Caldwell headed away from goal but it fell invitingly for Koren and the Slovenian international rifled a blistering volley into the top corner.
Only the woodwork stopped Blake restoring Burnley's lead after his powerful left-footed drive thumped against the crossbar.
Four minutes into first-half injury-time Kim put the Baggies ahead with a 20-yard deflected free-kick after he had been fouled by Clarke Carlisle which wrong footed goalkeeper Brian Jensen.
In a dramatic second half both sides created chances and West Brom could have wrapped the game up when Roman Bednar burst past Michael Duff, rounded Jensen but he could not find the finish as his shot rebounded off the post.
Burnley upped the tempo late on and West Brom had Carson to thanks for keeping the home side's lead intact as he saved acrobatically from Steve Thompson's drive.
With two minutes to play Burnley forced the replay that their endeavour deserved.
Blake found Elliott and the Burnley winger ran at the West Brom defence driving over a low cross which found Paterson who poked home.
West Brom manager Tony Mowbray:
"I was disappointed with the performance in the first half. You have got to give Burnley a lot of credit. They play the game in a manner I appreciate.
"Having got in front at half-time, we should have gone on to finish the game off and we had enough opportunities after half-time to do that.
"At 3-1 it would have been a long way back for them after all the effort they put in this week.
"They always had something to chase and go for and they got a goal and good luck to them. They took a wee chance by pushing extra men forward and they reaped the rewards."
Burnley manager Owen Coyle:
"Everyone expected us to turn up here and roll over but everyone who knows me will know I have unbelievable faith in this set of lads.
"We have gone 2-1 behind right on half-time and we have come in and it would have been easy to feel sorry for ourselves.
"But I have so much pride at the way they went about things, and the character and the desire they showed for the football club and for each other, and that is very important."
West Brom: Carson, Hoefkens, Pele, Donk (Zuiverloon 72), Robinson, Kim, Greening (Borja Valero 40), Filipe Teixeira (Brunt 71), Koren, Simpson, Bednar.
Subs Not Used: Kiely, Cech, Dorrans, Fortune.
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