A worldie from Cameron Archer looked like it would secure The Blades their first win back in the top flight but they were dealt a hammer blow by second-half substitute Jean-Ricner Bellegarde, who slotted home the equaliser in the 89th minute.
In a last ditch throw of the dice, The Blades were awarded a controversial penalty, which Norwood blasted into the back of the net to lift to claim a much-needed three points for his side.
Sheffield United came into this match still in search of their first league win this season against a Wolverhampton Wanderers side that were unbeaten in four matches, which included a win over Manchester City and a draw against Newcastle United.
An under-pressure Paul Heckingbottom mentioned before the match that while he is in regular contact with the club's board and feels their support, he does accept that results dictate the job.
Wolves looked the brighter of the sides after a stop-start opening to the match and got the first attack of the game, but Matheus Cunha fired his shot wide in wet conditions at Bramall Lane.
The Blades settled into the game as the first half went on, but Wolves were the side on top and almost took the lead before the half-hour mark when Hwang Hee-chan slid a pass into Cunha, who forced a good save from Wes Foderingham high to his right.
As half-time approached, Sheffield felt they should have had a penalty when Rhian Brewster went down in the box during a tussle with Toti Gomes, but the referee waved it away.
Brewster, who was only playing his second match since returning from a spell on the sidelines, was withdrawn moments later, holding the back of his leg, and Heckingbottom will be hoping that the 23-year-old is not the latest addition to his already fairly extensive injury list.
Clear-cut chances were at a premium in the first 45, and while neither side fully clicked going forward, the visitors did carry the greater threat and bossed possession in the first period.
As the rain got heavier during the break, it was the hosts that came flying out blocks in the second half, with Archer firing a shot just over within moments of the restart, while shortly after that, James McAtee put a free-kick from just outside the box narrowly wide.
The Blades were very good from then on, piling the pressure on the Wolves defence and they finally got the breakthrough they longed for when Archer broke clear of the defence and unleashed a rocket off the underside of the crossbar and into the back of the net.
It looked like Sheffield were home and dry, but that was until Bellegarde latched onto a poor clearance and stuck his effort in the top corner in the final minute of regulation time.
But, it was not over as the hosts threw everything forward in stoppage time and were awarded a late penalty after a VAR review, when Fabio Silva brought down George Baldock in the box.
Norwood stepped up to take the spot-kick and rifled it home to the elation of the home fans.
A huge result for Heckingbottom, who will need to reset for a clash against Brighton & Hove Albion next week, while Wolves will be playing host to Tottenham Hotspur.