Portuguese forward Matias was controversially shown a straight red after tangling with Blues substitute Cohen Bramall while Pudil collected two bookings either side of half-time.
The Owls were run ragged defensively in a shambolic first half, with Jota (2) and David Davis registering at Hillsborough.
The hosts showed plenty of grit and determination after the restart and Lucas Joao pulled one back but Birmingham held on to to condemn Wednesday manager Jos Luhukay to his first defeat in charge.
The result means the Owls have recorded just three victories from their last 17 competitive matches.
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Luhukay, missing 10 first-team players through injury, shuffled his pack, handing recalls to captain Glenn Loovens, Matias and Atdhe Nuhiu. Jordan Thorniley, Ross Wallace and Lucas Joao dropped to the bench. Jordan Rhodes was a notable absentee from the matchday squad.
For our Owls news and opinion on twitter follow hereThe hosts made a bright start and Matias, fit-again following a thigh problem, drilled a right foot shot over before Adam Reach's fierce low left foot shot was well held by David Stockdale.
But out of nothing, Davis tapped Birmingham in front with eight minutes on the clock after a mix-up at the back between Frederico Venancio and Joe Wildsmith. Venancio tried to chest the ball back to the Owls shot-stopper, who came out to collect a cross, and Davis pounced to grab his second goal in as many matches.
Defensively Wednesday looked a little uncertain at times, making a number of uncharacteristic mistakes.
And Jota almost punished them after Morgan Fox cheaply gave the ball away but Daniel Pudil came to their rescue, producing two excellent blocks.
Spanish forward Jota, the Blues' record-signing, deservedly doubled the Blues' advantage midway through the half. The Spanish attacker tried his luck from long range and his left foot strike squirmed through Wildsmith's hands into the back of the net. It was a howler from Wildsmith but Jota was given far too much time and space to line up his shot.
At the other end, Pudil glanced Reach's free kick well wide before Luhukay made a tactical substitution. He took off Loovens and threw on Joao, switching from a 3-5-2 system to 4-3-3.
Birmingham were forced into making a change when Jonathan Grounds picked up an injury and had to be taken off on a stretcher. He was replaced by Arsenal loanee Cohen Bramall, who had a trial spell at Hillsborough last year.
For more news from Hillsborough click hereThings went from bad worse for the Owls in the 38th minute when Matias was dismissed after tangling with Bramall. The pair squared up to each other and referee Scott Duncan deemed Matias to have instigated the off the ball incident.
Birmingham continued to threaten and Jota's effort from an acute angle went into the side-netting after a good run and cross down the left by Bramall.
But Jota was not to be denied deep into added on time. He tapped in the Blues' third after Wildsmith made a smart stop to keep out Davis's attempt.
Boos greeted the half-time whistle and Wednesday's fans vented their fury at Duncan as the officials trudged back into the dressing room.
Joao reduced the arrears in the 54th minute. The Portugal international jinked his way into the penalty area before rifling a rasping left foot shot past Stockdale. It was Joao's third of the campaign and ended the Owls' long goal drought stretching back to Boxing Day.
Birmingham responded well and Wildsmith parried away Davis' first-team attempt away. Moments later, Sam Gallagher narrowly evaded Jota's delivery as the Blues pressed hard for a fourth.
It was a spirited second half response by the Owls and Stockdale had to be alert to keep out Jacob Butterfield's 25 yard shot.
Pudil, who was booked in the first half, also saw red after bringing down Maxime Collin near Wednesday's penalty area.
Joao worked tirelessly up front and he tricked his way past Marc Roberts in the 85th minute only to see his left foot piledriver fly agonisingly over the top.
At the other end, Wildsmith made a good stop at his near post to deny Jeremie Boga but it was an afternoon to forget for Wednesday.
Owls: Wildsmith; Venancio, Loovens (Joao 26), Pudil; Palmer, Butterfield (Thorniley 71), Pelupessy, Reach, Fox; Matias, Nuhiu (Boyd 72). Substitutes: Dawson, Hunt, Jones, Wallace.
Birmingham: Stockdale; Colin, Roberts, Dean, Grounds (Bramall 33); Kieftenbeld (Gardner 75), Davis (Lowe 84), Ndoye; Jota, Boga, Gallagher. Subs: Trueman, Morrison, Lowe, Adams, Jutkiewicz.