The Baggies, who are already promoted, travel to Selhurst Park on Monday to face Crystal Palace, the only side still within reach for the third-bottom Owls after Watford's 3-0 victory over Reading.
Should the Eagles triumph then Wednesday will join Plymouth and Peterborough in League One next season.
Any other result would give Alan Irvine's outfit one last crack at survival on the final weekend of the season when the two former established Premier League clubs go head to head at Hillsborough in a titanic showdown, although only victory will do.
It was the visitors who took the lead with just 15 minutes on the clock through Jermaine Johnson's low finish but their advantage lasted a mere 67 seconds as Peter Whittingham levelled with a sublime free-kick.
Jay Bothroyd then handed Cardiff the lead and, although Marcus Tudgay levelled matters 12 minutes from time, the former Wolves striker quickly broke hearts to restore City's advantage and all but secure home advantage in the second leg of the play-offs.
It was the Bluebirds who started on the front foot and Owls keeper Lee Grant was called upon to twice rescue his side in a matter of seconds from Bothroyd and Joe Ledley.
But Bluebirds keeper David Marshall soon proved he was equal to his opposite number with a great diving save of his own to tip Tommy Spurr's goalbound free-kick onto a post.
However the 2,000 or so travelling Wednesday fans did not have to wait long for a breakthrough.
Tom Soares' ball found Johnson and he hammered a low drive into the bottom corner from six yards out, sending the visiting support wild.
Yet their delight was short-lived as Whittingham left Grant stranded with a delightful free-kick from the edge of the area after being brought down by former Cardiff captain Darren Purse - his 20th league goal of the season.
The drama of the opening 20 minutes cooled somewhat as the first half wore on and, with Wednesday desperately needing a big second-half display, it was Cardiff who took the lead nine minutes after the restart.
Stephen McPhail's ball straight over the top split a Wednesday defence that was all over the place and Bothroyd coolly brought the ball down before slotting past Grant.
Play went from end to end during a second half in which the visitors saw their survival hopes slowly ebb away, that was until Tudgay threw them a lifeline in the 78th minute.
Despite the leveller, the south Yorkshire outfit desperately needed all three points and their hopes were extinguished within a matter of minutes as the impressive Bothroyd registered his second of the afternoon.
Etienne Esajas hit the inside of a post in injury time but Cardiff held on to push Wednesday ever closer to the drop and gain revenge for their 1-0 defeat at Hillsborough on the final day of last season which ended their own play-off aspirations.