In a fluctuating encounter Martin Scott's Pools side were outplayed in the opening 45 minutes only to hit back to lead and come within nine minutes of clinching promotion.
The Owls dominated the first half - they settled quickly and kept Pools penned back in their own half for long periods.
Lee Peacock headed James Quinn's tenth minute cross wide before Peacock shot over the bar two minutes later. Chris Brunt then saw his shot saved by Dimitrios Konstantopoulos as Sturrock's side started strongly.
Craig Rocastle dragged a 17th minute effort wide and Peacock was just unable to latch on to Jon-Paul McGovern's cross while Brunt fired a 33rd minute shot over the bar.
Pools' only real opportunities of the first half came late on when Joel Porter's lob was just too high and Adam Boyd mishit his shot when well-placed.
Back came Wednesday though and Peacock should have done better when he fired over the bar after creating a clear opening for himself.
Just when it looked as though the first half was going to end goalless, McGovern finally broke the deadlock in stoppage time. A neat one-two between Rocastle and Peacock allowed Rocastle to get to the byline and, when he pulled the ball back, McGovern beat two Pools defenders to the ball before lashing a shot into the roof of the net.
However, within two minutes of the restart a long throw flicked off the head of Bullen allowing Pools substitute Eifion Williams to run onto the ball and fire home.
Hartlepool were much brighter in the second half and on the hour Anthony Sweeney fired wide after a probing passing movement by Scott's side. Three minutes later Wednesday's McGovern fired into the side-netting and in the 66th minute Brunt drifted a free-kick just off target.
In the 71st minute Hartlepool took the lead when another substitute, Jon Daly, powered a header home from Gavin Strachan's precise free-kick after only coming on to the pitch 60 seconds earlier after replacing the injured Porter.
It looked as though the underdogs were going to have their day but, with just nine minutes left, Chris Westwood was adjudged to have pulled Drew Talbot back, and after the Pools defender was sent off, Wednesday substitute Steve MacLean scored from the spot.
The game swung in the Owls' favour and in the fourth minute of extra-time a mistake by Michael Nelson allowed man-of-the-match Glenn Whelan to burst clear and steady himself before firing home.
After a few anxious moments, teenager Talbot beat the offside trap in the last minute of extra time, rounded Konstantopoulos and coolly slotted the ball into the empty net to guarantee Championship football for Sturrock's side next season.