Albion, who had striker Jay Rodriguez sent-off in the 63rd minute for appearing to stamp on Marco Matias, were heading for defeat after Atdhe Nuhiu had given Wednesday a fifth minute lead.
But Brunt bundled home a Matt Phillips corner deep into stoppage time to extend promotion chasing Albion's unbeaten run to nine games.
Nuhiu's strike was only his second for Wednesday this season and his first since the opening day of the campaign when he was sent-off against Wigan.
It looked to be good enough to earn Wednesday a third straight win under the control of caretaker boss Lee Bullen before Brunt's late heroics earned Albion a vital point in their quest for promotion.
Wednesday shocked West Brom by taking the lead in their very first attack of note.
After forcing a left-wing corner, Adam Reach floated over a cross that captain Tom Lees nodded back across the face of goal.
Nuhiu reacted quicker than any of the Albion defenders and stabbed the ball past goalkeeper Sam Johnstone from close range.
The hosts were rocked back on their heels by the shock of falling behind and struggled to threaten to find an equaliser.
The closest that they came in the opening 20 minutes was when on-loan Leicester winger Harvey Barnes cut in from the right flank and blazed his shot well over the bar.
In fact, Albion could easily have fallen further in the 26th minute only to be saved by Johnstone.
Wednesday midfielder Barry Bannan floated a perfectly placed pass over the top of the Albion defence which picked out the on-rushing Matias, whose low cross shot was tipped away at full stretch by former Manchester United 'keeper Johnstone.
Johnstone then came to Albion's rescue again five minutes later when Wednesday central defender Michael Hector played the ball to Morgan Fox at the far post and his shot was this time blocked to safety.
After riding their luck, Albion did start the second-half on the front foot but crosses from skipper Brunt and Barnes ultimately came to nothing.
Albion's task then became a whole lot harder when Rodriguez was shown a straight red card by referee Andy Davies after tangling with Matias.
Ironically, only moments after the dismissal of Rodriguez, Albion created their best chance of the game in the 68th minute when Kieran Gibbs crossed for substitute Hal Robson-Kanu whose header was spectacularly tipped away by Wednesday goalkeeper Cameron Dawson.
But their was nothing that Dawson could do to deny Brunt in a dramatic finale.