Goals from Glenn Whelan and Madjid Bougherra in a five-minute lightening spell tore apart the Baggies, who have now lost six of their last nine games.
Mowbray's £22million side, expected to make a quick return to the top-flight, were second best to Brian Laws' cut-price outfit who are on a winning streak with just one defeat in eight games.
Winger Chris Brunt, back after a one-match suspension, set up the Hillsborough side's opener in the tenth minute running on to Deon Burton's pass to pump in a curling cross from the left.
Whelan beat his marker Paul Robinson at the far post to head past the helpless Russell Hoult from five yards.
Whelan almost snatched a second with a cheeky lob from 25 yards but the ball looped the wrong side of the bar as Hoult scrambled back to cover his line.
Wednesday heaped more misery on the Midlands side in the 15th minute. Kenny Lunt's inch-perfect free-kick found centre-back Bougherra who powered the ball over the line from close range for his second of the season.
West Brom, closed down by the Wednesday midfield, struggled to find an opening and Kevin Phillips ballooned a right-foot strike high over the bar in frustration from 20 yards.
Wayne Andrews, on loan from Coventry, tested Hoult with a low effort and almost caught him out as he chased a long ball downfield but the Baggies keeper raced out of his box to hack the ball away.
Dangerman Andrews had the West Brom defence back pedaling minutes later as he surged towards the area before drilling in a low strike but the ball flashed a fraction wide of the far post.
Wednesday pushed for a third and Hoult, under pressure from Marcus Tudgay, failed to hold on to Brunt's teasing corner but the ball was hacked away before the little striker could add the finishing touch.
Senegal international Diomansy Kamara almost found the target as he stretched to meet a Nigel Quashie corner and beat his marker but headed inches wide from six yards.
A rare Bougherra slip up let in Phillips in the 82nd minute but his low shot was pushed round the post in spectacular fashion by on loan keeper Mark Crossley.
Jason Koumas pulled one back for West Brom in the 89th minute before substitute Steven MacLean added Wednesday's third from the penalty spot in injury time after he had been upended by Jonathan Greening.