It was looking as though gutsy Town were set to all-but seal their third tier safety as they led 2-1 with 17 minutes to go but two clinical strikes inside 45 seconds from Lee Gregory – to complete a brilliant hat-trick – handed Wednesday the spoils.
Town must now head to Bolton for their final-day decider on Saturday with their fate far from secure as they remain outside the bottom four on goal difference only.
Ahead of last night’s penultimate League One game of the season at a sun-drenched Highbury Stadium, Town boss Stephen Crainey had tweaked his starting line-up.
The Scot made three changes following the weekend draw against drop rivals Wimbledon, with one enforced as Saturday’s point-saver Joe Garner came in up top for the injured Ellis Harrison.
Dylan Boyle and Tom Clarke stepped up for starts, with Josh Harrop and Anthony Pilkington dropping to the bench.
There was a cracking atmosphere at kick-off time, with the Cod Army in full voice and almost 2,000 Wednesday fans bidding to roar their team into a play-off position.
Unfortunately for Town, it was the Owls’ travelling army who were celebrating wildly after only three minutes.
Experienced hitman Gregory slotted Wednesday in front after he’d latched on to Jack Hunt’s through-ball.
The Owls dominated but somehow the hosts levelled after 18 minutes.
Cian Hayes’ low cross wasn’t cleared by a hesitant Wednesday defence, and Callum Camps was on hand to smash home the loose ball from the edge of the box.
The equaliser temporarily knocked the stuffing out of Wednesday and with Town now re-energised, Dan Batty fired in a shot which was blocked by Sam Hutchinson.
Town stormed into the lead 11 minutes before half-time, Garner darting in to thump home a header from Batty’s measured free-kick.
The Cod Army fans celebrated wildly but the woodwork saved their team twice within a matter of seconds.
The excellent Barry Bannan clipped a terrific effort against a post before Callum Paterson’s shot from the rebound deflected against the opposite post.
Wednesday sub Josh Windass side-footed a decent opportunity over the top after being teed up neatly by Bannan just past the hour.
The Owls were once again camped in the Town half but a vital breakthrough eluded them until Gregory’s lethal double intervention in the 73rd and 74th minutes.
He nodded home a sweet cross from Massimo Luongo and sent the travelling faithful wild as he robbed Batty before drilling past Cairns from just outside the box.
The score remained the same, which was harsh on a Town side who had gone toe-to-toe with moneybags Wednesday.
The win lifts the Owls three places to fourth, while Town's destiny is still in their own hands, though another slip-up at Bolton could be fatal.
Fleetwood: Cairns, Clarke, Jules (Harrop 86), Nsiala, Andrew, MacAdam, Camps, Batty (Pilkington 74), Boyle, Hayes, Garner; Subs not used: O’Hara, Butterworth, Biggins, Baggley, Bird.
Sheff Wed: Peacock-Farrell, Hunt, Storey, Hutchinson, Palmer, Johnson (Windass 62), Bannan, Luongo, Paterson, Byers (Mendez-Laing 62), Gregory (Dunkley 89); Subs not used: Wildsmith, Gibson, Kamberi, Berahino.
Referee: Carl Boyeson; Att: 3,901