Nuno Espirito Santo's charges were handed promotion after Fulham slipped up against Brentford on Saturday, leaving Wolves to cap off a memorable weekend by earning a dominant win thanks to goals from Diogo Jota and Benik Afobe in the derby.
Wolves started the game with typical possessional dominance, but the hosts fell asleep at the back in the 11th minute, allowing Lukas Jutkiewicz to break in behind before being thwarted by two standout saves from John Ruddy.
Those early chances were the best of the half for Birmingham, who despite pressing with impressive energy and intent, found themselves passed into relative submission by a dominant Wolves outfit.
It took the home side the best part of 21 minutes to carve out a clear-cut opportunity, but they made the most of it to take the lead.
Helder Costa impressed on the right flank and he was crucial as Wanderers went in search of a second, the winger beating his marker before squaring into the middle for substitute Leo Bonatini, who could only thump an effort into the grateful arms of Stockdale.
Any hope of a City turnaround was scuppered early on after the interval, when Costa drew a sliding challenge from Harlee Dean on the edge of the box, who was then dismissed with a straight red having been deemed the last man in defence.
The second period was certainly more attritional than the first and Wolves might have just put the game to bed with 20 minutes remaining, although Stockdale did well to save from Romain Saiss after he broke clear from Marc Roberts.
Birmingham continued to scrap and Jutkiewicz passed up a half-chance in the closing stages, before City were punished by some clinical work from the hosts, Alfred N'Diaye threading Afobe in behind, who chipped the ball over Stockdale to seal all three points.
Victory for Wanderers now leaves them within touching distance of Championship glory, while defeat for Garry Monk's men pushes Birmingham right into the midst of a developing relegation scrap.
WOLVERHAMPTON WANDERERS (3-4-3): Ruddy; Bennett; Coady, Boly; Doherty, Neves (N'Diaye 70'), Saiss, Douglas; Costa, Jota (Afobe 74'), Cavaleiro (Bonatini 16')
BIRMINGHAM CITY (4-5-1): Stockdale; Harding, Morrison, Dean, Colin; Grounds (Maghoma 62'), Kieftenbeld (Roberts 54'), Ndoye (Gardner 78'), Davis, Jota; Jutkiewicz