It was a poor display from the Owls, who found themselves two-nil down at the break thanks to a header from Mark Shelton and a comical own goal from Jaden Brown.
Things did not improve in the second half for Wednesday, though, as Will Goodwin confirmed their passage into the next round after the hour mark.
Story of the match
It was a sombre start to the game, with neither side being able to create anything clear-cut, but the away side arguably edged it as Wednesday made several comical errors on the evening, which frustrated the Hillsborough crowd.
The Monkey Hangers took the lead on 12 minutes through Shelton, who rose highest at the back post to connect with Tom Crawford’s floated delivery and head home his first goal of the season.
Things got worse for the Owls just two minutes later when Brown placed a careless back pass past Joe Wildsmith, who had pulled away from his goal-line to make an option, which doubled the League Two side's lead.
Wednesday almost halved the deficit instantly, with the ever-bright Corbeanu jinking past several players on his way into the area. But he saw his shot turned behind by Jonathan Mitchell.
The Owls upped the ante after a horrendous two-minute spell; however, they were poor and sloppy in their endeavours.
On the half-hour mark, Wednesday managed to break through on goal as Sylla Sow and Saido Berahino linked up on halfway, allowing the former to close down on goal but Timi Odusina raced back and dispossessed the forward.
Despite several half-chances for the Owls, the visitors refused to sit back as they ventured forward and caused Wednesday some more problems.
Joe Grey saw a long-range effort pushed behind by Wildsmith before Michael Smith fired wide from Jamie Sterry’s cross.
Wednesday ought to have pulled a goal back just before the break, as a Corbeanu set-piece landed at the feet of an unmarked George Byers, but the midfielder snatched at it and put it wide of the far post.
Dele-Bashiru had the second period’s first chance, but the midfielder’s long-range effort went high and wide of the target.
However, it was much of the same from Wednesday as the home crowd continued to groan in frustration after another poor showing.
On the hour, Berahino went close to pulling a goal back. Bannan carried the ball down the left before he slipped in Corbeanu, whose cross was stabbed towards goal by the forward but held by Mitchell.
However, like the opening goal, Hartlepool survived that early pressure to score again from a corner, with Goodwin poking a loose ball past Wildsmith in the area.
On 66 minutes, Corbeanu sent a teasing ball towards the back stick where Hunt lost his marker but failed to get a toe-end on the delivery.
Possession ebbed and flowed between the two sides before Wednesday wasted another chance with seven minutes remaining.
Callum Paterson controlled a cross into the area but was tackled by Eddie Jones. However, the defender was unable to get the ball out of his feet and Windass reclaimed possession before he struck a shot over the bar.
Hartlepool saw out the rest of the game to progress in the competition.
Takeaways from the game
A happy beginning for the new manager
Newly appointed Pools boss Graeme Lee watched his new side after being appointed on the morning of the fixture.
He will have been impressed with what he saw from Hartlepool, as they put three past Lee’s former club.
The Monkey Hangers were resolutely strong at the back, soaking up forward pressure from Wednesday before hitting them on the counterattack.
The 43-year-old will then look to break a run of five straight league defeats on Wednesday night when Pools entertain Rochdale at Victoria Park.
Peripheral players fail to shine
Wednesday made eight changes from the side that drew 2-2 with Wycombe Wanderers, with players such as Sow, Berahino, and Brown being given an opportunity to force their way back into Moore’s plans ahead of a midweek trip to Portsmouth in six days.
However, it was a largely uninspiring performance from those who had slipped down the pecking order as the Owls bowed out of the EFL Trophy after winning all three group games.
Only Corbeanu arguably stood out for Wednesday, as the winger tried to create things going forward as he was progressive in possession. The Owls, however, were not as their negative, sloppy play was blasted by the home support.