Steve Stone's U23 side deservedly collected their second point in Group C of the Premier League Cup as a late Dan Agyei equaliser extended their unbeaten run to eight game.
The Clarets followed up a draw at Liverpool with another impressive performance against higher-ranked opposition to keep their qualification hopes alive.
Liverpool's 1-1 draw with Bournemouth later in the weekend sent Burnley back to the bottom of the group, but with three home games to come their destiny remains in their own hands.
West Brom, a category-one Academy outfit who play in Premier League 2, fielded a side that hosted over 500 professional appearances between them in Bakary Sako, former Claret Tyrone Mears, Oliver Burke and Sam Field.
Meanwhile, the Clarets handed a first start to 17 year-old Richard Taylor, who played 67 minutes of his under-23 debut.
Burnley took the lead with 25 minutes played, as Marley Blair curled home a delightful effort.
However, the Baggies levelled just minutes later through Rakeem Harper.
And it was the home side that struck again in the 65th minute through Daniel Meredith.
Stone's side then snatched a deserved equaliser 10 minutes later through Dan Agyei.
The match took a while to get going at Hednesford Town's Keys Park stadium, the first real chance coming when Agyei's header was hacked away off the line by defender Nathan Ferguson.
And the home side came close shortly after, as Harper's volley sailed just wide of the post.
However, it was the Clarets that took the lead just after, as Dwight McNeil picked out Blair with a wonderful pass out on the left, before the winger cut inside and curled the ball into the top corner.
The lead didn't last for long though, as a corner fell to Harper at the back post, who stabbed home the equaliser just before the 30 minute mark.
And the Baggies came close to taking the lead just before the break, as Bakary Sako cut the ball back to Oliver Burke, however, the former Red Bull Leizpig winger's effort was excellently denied by Adam Legzdins.
Burnley started the second-half bright and Josh Benson came close to finding the back of the net early on, as his ambitious strike from range floated just over the bar.
However, it was West Brom that struck next, as Burke teed up Meredith from close range to tap home past Legzdins against the run of play.
The Clarets continued to dominate the majority of the ball and got their equaliser 10 minutes later.
Substitute Mark Howarth played a neat ball through into the path of Agyei, who coolly finished into the bottom corner.
Burnley: Legzdins, Cropper, Clarke, Glennon, Koiki, Benson, Taylor (Howarth 67), McNeil, Bayode, Agyei, Blair (Harker 90). Substitutes: McMahon, Cook, Howarth, Harker, Thomas.
West Brom: Palmer, Mears, Ferguson, Howkins, Townsend, Meredith, Field, Harper, Leko, Burke, Sako. Substitutes: Keranovic, Healy, Wilding, Azaz, Rogers.