For the majority of the first half Peterborough had to thwart a barrage of attacks from Wednesday, with visiting goalkeeper Joe Lewis saving well from Leon Clarke's powerful volley before Darren Potter saw his 20-yard drive rebound back off the crossbar.
However, the visitors' resistance eventually buckled on the stroke of half-time when Tudgay got ahead of his defender to nestle home a glancing header from Michael Gray's corner.
Peterborough were sent out early after the break in an attempt to shock them into a performance. But it did not have the desired effect as Tudgay soon doubled the home side's advantage with another well-timed header.
Jake Livermore pulled one back for Peterborough with 15 minutes remaining to set up a nervy finish, but it proved a mere consolation as Wednesday weathered the late storm to secure all three points.
A bumper crowd of 24,882 were in attendance on a grey afternoon at Hillsborough, spurred on by Wednesday's 'kid for a quid' deal and the chance to see their new manager first hand.
The Owls, who have been in imperious form since the former Preston boss took charge, began brightly.
The home side's first chance came after just seven minutes when Clarke forced Lewis to save his well-struck drive.
Five minutes later, Tom Soares saw his drilled right-footed shot well blocked as Wednesday continued to enjoy the better of the opening exchanges.
The visitors had their first half chance in the 18th minute, when Reuben Reid got a header on target after getting on the end of Exodus Geohaghon's gargantuan throw-in.
But it proved a brief respite, as Wednesday continued to put Peterborough under the cosh, with Darren Potter smashing the crossbar from 20 yards before Charlie Lee's block prevented Tudgay from tucking in the rebound.
Wednesday continued to press forward and, after Darren Purse saw his far-post header palmed over, the hosts got the goal their dominance deserved.
Gray drilled a corner into the bustling six-yard box and Tudgay rose highest to send the ball into the back of the net and give his side a lead heading into the break.
Mark Cooper sent his team out early for the second half but they continued in the same vein, with Wednesday pressing them.
The Owls came agonisingly close to doubling their advantage in the 56th minute but Clarke, Tudgay and Soares were all blocked in quick succession by some last-ditch defending.
Wednesday continued to pile forward and, after skipper Purse put a header just over, Tudgay netted his second of the match, stooping in unmarked at the back post to extend the Owls' lead in the 67th minute.
Despite Wednesday looking in control, the visitors began to get a foothold back in the game and, after a sustained spell of pressure, managed to pull a goal back as Livermore found the bottom-left corner.
The Posh pressed incessantly in search of the equaliser but it proved elusive as Wednesday held on for the win.