The managerless Owls, still under caretaker boss Lee Bullen, led for most of the game thanksÂ to Atdhe Nuhiu's early strike.
Wednesday's defenceÂ withstood plenty of pressure throughout despite the second half dismissal of home striker Jay Rodriguez.
But they couldn't hold out for the three points as ex-Owl Chris Brunt stole a share of the spoils when he bundled in from a corner to break Wednesday hearts.
The Owls made three changes from the Middlesbrough win on Boxing Day with Keiren Westwood and Lucas Joao dropping out of the squad and Steven Fletcher dropping to the bench.
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In came Cameron Dawson, George Boyd and Nuhiu.
The latter was to make a rip-roaring start when he gave the Owls a fifth minute lead.
A corner fell his way and the Kosovo striker smashed the ball home from a couple of yards out.
The Baggies couldn't live with Wednesday at times, with Morgan Fox and Marco Matias both seeing efforts kept out in the first half hour.
The hosts came out of the blocks flying after the interval but Dawson did well to keep them at bay before Albion were reduced to ten men just after the hour.
Rodriguez was shown a straight red card for a coming-together with Marco Matias.
Albion boss Darren Moore immediately made three changes and one of those, Hal Robson-Kanu, almost restored parity.
The Wales striker's header was heading for the top corner before Dawson made a superb save to claw it behind for a corner.
Despite having the man advantage, Wednesday had to soak up plenty of pressure.
Just as it looked as though they would hold out for a third win on the trot, Brunt popped up in stoppage time and prodded home to seal a draw.
WBA: Johnston, Gibbs, Bartley, Adarabioyo (Phillips 65), Dawson, Livermore, Morrison (Barry 65), Brunt, Sako (Robson-Kanu 65), Barnes, Rodriguez
Subs not used: Myhill, Harper, Mears, Hoolahan
Owls: Dawson, Palmer, Lees, Hector, Fox, Bannan, Hutchinson (Jones 74), Reach (Pelupessy 36) Matias, Boyd, Nuhiu (Winnall 80)
Subs not used: Wildsmith, Baker, Thorniley, Fletcher