Jordan Rhodes gave the visitors the lead after 19 minutes as Wednesday looked to end a run of four consecutive draws, but second-half goals from James Maddison and Timm Klose turned the game in Norwich's favour.
The hosts then sealed the result with a penalty deep in stoppage time as Nelson Oliveira converted from the spot, handing Norwich their first victory since October and arresting their slide down the table.
Both sides were limited to long-range efforts in the opening stages of the contest, but Rhodes took full advantage of the first clear chance when he sent a stooping header from Adam Reach's cross past Angus Gunn.
Norwich had a big chance to respond with an equaliser shortly before the half-hour mark when Maddison slipped a pass through for Oliveira, but the Portuguese forward sent his finish wide of the target when he should have done better.
Oliveira came close again 10 minutes later when his header was pushed away by Kieren Westwood, although the hosts could have found themselves two goals down at the break but for some smart goalkeeping from Gunn just before half time.
The Canaries were the quickest out of the blocks for the second half too, and in a sign of things to come they threatened twice in the opening exchanges as first Oliveira saw his header tipped over before Mario Vrancic could only pick out the side-netting with his effort from outside the box.
The equalising goal did arrive 10 minutes after the restart, though, with Maddison lashing a rebound home after Alex Pritchard's initial effort had been parried by Westwood.
Norwich then completely turned the game around by taking the lead with just over 20 minutes remaining as Klose powered a towering header past the keeper from a corner to set his side on course for the overdue victory.
Reed was brought to the ground inside the penalty area at the end of a quick Norwich counter-attack, and Oliveira made no mistake from the resulting spot kick to wrap up the win and lift Norwich up to 15th, now 11 points clear of the relegation zone.
Wednesday are now level on points with the Canaries as their playoff hopes took another blow, with the result seeing them sink into the bottom half of the table and leaving them now 10 points off the top six.
Norwich City (4-2-3-1): Gunn; Pinto, Hanley, Klose, Stiepermann; Reed, Vrancic, Murphy (Watkins, 46'), Maddison (Zimmerman, 92'), Pritchard (Trybull, 79'); Oliveira
Sheffield Wednesday (4-4-2): Westwood; Palmer, Loovens, Van Aken, Fox (Nuhiu, 78'); Wallace, Jones (Lee, 69'), Bannan, Reach; Rhodes, Hooper