The Owls had not lost at The Hawthorns since 1975 and they had plenty of chances to maintain that record, but Baggies substitute Robert Koren made them pay with a stunning long-range strike 12 minutes from time.
The game started brightly, with West Brom dominating possession but Wednesday hitting them on the counter-attack, Jermaine Johnson jinking past three defenders before firing in a shot that was comfortable for Scott Carson.
The Baggies' response came from Chris Brunt after good interplay from Graham Dorrans, Jerome Thomas and Marek Cech, but his overhead kick whistled inches wide.
The home side were grateful to defender Gabriel Tamas in the 33rd minute when the lively Johnson went past Steven Reid, got to the byline and cut the ball back for Luke Varney who looked set to slot home before Tamas' superb last-ditch challenge.
It was edgy times for Roberto Di Matteo's men as half-time approached, with the visitors breaking quickly again from a West Brom corner.
Varney found himself one-on-one with Carson, but he tried to be too clever and his chipped attempt cleared the crossbar.
The Owls came close again at the start of the second half from a well-worked set-piece. Varney cut the ball back for James O'Connor and he laid the ball off for Leon Clarke, but the striker blazed over.
Ishmael Miller came on for the ineffective Luke Moore and the bustling forward was straight into the action, barging his way past two defenders before firing inches past the post.
Miller had changed the game in the Baggies' favour, and he should have done better with another effort shortly afterwards after Jonas Olsson had seen a header parried by Grant.
It was all West Brom, and after Miller brought a superb save from Grant with a close-range header in the 65th minute, the keeper pulled off an even better save to deny James Morrison from point-blank range from the resulting corner.
Wednesday hit back as Clarke charged through the home defence and lashed in a shot which Carson did well to palm away.
But it was West Brom who looked the most likely, Dorrans letting fly from 30 yards with the keeper scrambling to keep it out.
A goal finally arrived in the 78th minute from a superb passing move. Morrison played the ball to Dorrans who in turn fed Koren 30 yards out and he lashed the ball past Lee Grant for his fifth goal of the season.
Koren almost added a second at the death, but his 12-yard shot flew wide.