Goals from Ashley Barnes and Danny Ings put the Clarets into an early two-goal lead, but Chris Brunt and Brown Ideye ensured a share of the spoils
First-half headers from Ashley Barnes and Danny Ings put the home side firmly in control, until Chris Brunt reduced the arrears from a corner on the stroke of half-time.
And it was another set-piece that proved Burnley's downfall midway through the second period as substitute Ideye glanced home Brunt's delivery.
For West Brom, captained by Darren Fletcher on debut following a move from Manchester United, it was a positive response to last week's 3-0 defeat by Tottenham, though Pulis benched Saido Berahino following comments in the week about his future at the club.
The point takes Burnley back out of the bottom three after Hull City had displaced them on Saturday but with a tough run of fixtures to come, Dyche will see this as a huge opportunity lost.
Burnley, who named the same starting XI for the 10th league game in succession, had the better of the opening exchanges with Ings firing over before Ben Foster produced a superb one-handed save to tip Scott Arfield's effort onto the crossbar.
And Burnley's pressure paid off in the 11th minute as they broke the deadlock.
Arfield headed the ball into the path of Ings, and his hooked cross found strike partner Barnes who nodded in from six yards.
Berahino, who attracted attention in midweek when he suggested he could move away from West Brom at the end of the season, was soon summoned into the action due to an injury to Victor Anichebe.
The visitors managed to get something of a foothold in the game, but Burnley doubled their lead in the 32nd minute thanks to their top scorer.
A clever dummy from Kieran Trippier down the right-hand side released George Boyd, whose inviting cross was met by a powerful Ings header, leaving Foster stranded.
Just when it seemed the hosts would reach the interval with a comfortable lead, Brunt lost his man at a corner and headed Callum McManaman's delivery in off the bar from close range.
Brunt then turned provider for the equaliser, sending in a corner from the right that half-time substitute Ideye flicked home having escaped the attentions of David Jones.
Joleon Lescott headed Jason Shackell's effort off the line, while Gareth McAuley was fortunate to see a handball in his own penalty area go unpunished.
Berahino was almost the hero for Albion in the dying moments, but Tom Heaton and Ben Mee combined to keep him out and prevent Burnley from capitulating fully on home turf again.