The centre-back, on loan from Premier League Watford, volleyed home Matt Grimes' corner deep into added on time to extend Wednesday's winless run to three matches.
The Owls trailed at half-time after a poacher's goal by Andre Ayew.
But Monk's men looked set to return to winning ways after goals from substitute Fernando Forestieri and Morgan Fox.
Wilmot, though, levelled to ensure the spoils were share in a thrilling clash.
A point was the least Wednesday deserved after dominating the Championship duel for long periods. Monk’s side hit the woodwork three times either side of half-time and the result keeps them in seventh position going into the international break.
After an impeccable minute's silence and rendition of the Last Post to mark Remembrance Sunday, Wednesday made a strong start.
It took a desperate piece of defending from Wilmot to prevent recalled striker Atdhe Nuhiu from getting on the end of Kieran Lee's delivery from the right.
A corner routine straight from the training ground almost caught the Swans out in the fourth minute. Kadeem Harris showed good vision to find Jacob Murphy, who was preferred in the wide role to Adam Reach, but the Newcastle United loanee's right foot drive struck the post.
There were concerns over the fixture going ahead after flooding hit the Northern part of England on Thursday but the playing surface was in immaculate condition.
And Wednesday enjoyed the lion-share of possession and territory early on as Nuhiu's right foot curler from 25 yards flew inches wide.
The hosts were causing problems from set plays, with Steven Fletcher denied by Woodman after a dangerous Barry Bannan corner. Fletcher hooked the ball back into the six-yard box only for Nuhiu's header to be deflected over the top.
Against the run of play, Swansea threatened to break the deadlock in the 17th minute when Ayew collected Bersant Celina's pass but Liam Palmer came to the Owls' rescue by booting the ball away to safety.
Keiren Westwood spared Iorfa's blushes in the 25th minute, making a routine stop after the ball deflected off the defender towards his own goal.
Wednesday came within a whisker of taking the lead on the half hour mark. Murphy's wicked corner to the near post somehow evaded Nuhiu and Fletcher but the Owls recycled possession and Fletcher saw his downward header from Palmer's cross hit the left hand upright. Iorfa's follow up shot was blocked by a last-ditch tackle as the Owls continued to pile on the pressure.
But Swansea punished Monk's side for their wayward finishing in the 32nd minute when Andre Ayew poked them in front after Westwood fumbled Matt Grimes' corner and Morgan Fox failed to clear his lines. Ayew was the quickest react to claim his sixth goal of the season.
Bannan, restored to the centre after recovering from an ankle problem, tried his luck from long range in the dying embers of the first half but Woodman was equal to his strike.
The Owls created the first meaningful opportunity after the break as Murphy prodded Fletcher's inviting delivery from the left just wide.
Chances continued to come and go for Wednesday, with Fletcher's header from Kadeem Harris's delivery blocked by Woodman before Murphy's follow up clattered the post.
Nuhiu had a strong penalty shout rejected by referee David Webb after the giant forward appeared to be pulled down by Jake Bidwell. The decision incensed the home faithful.
But Wednesday were fortunate not go two behind after Wayne Routledge dinked a glorious cross to the back post towards Ayew but the striker nodded wide off balance.
Fox produced a vital block to prevent Dyer from forcing Westwood into action following a fine raid down the left by Bidwell as the Swans pushed hard for a second.
With 11 minutes remaining, Westwood redeemed himself by pulling off an outstanding save to keep out Ayew's shot on the turn.
It proved a big moment as Forestieri, a second half replacement for Nuhiu, turned in Lee's delightful pull back from the byline after Woodman superbly saved Adam Reach's rasping low drive.
Fox completed a remarkable turnaround by firing home from close range after Swansea failed to deal with a corner to spark wild celebration scenes among the home contingent.
It looked like Wednesday would hold out for maximum points after Westwood produced one of the saves of the season to keep Bryers at bay.
But Wilmot had the final word, turning home from the resulting corner to ensure it finished all square.
Owls: Westwood; Palmer, Lees, Iorfa, Fox; Murphy (Reach 76), Bannan, Lee, Harris (Rhodes 83); Fletcher, Nuhiu (Forestieri 66). Substitutes: Dawson, Pelupessy, Thorniley, Odubajo.
Swansea: Woodman; Naughton, Wilmot, van der Hoorn, Bidwell; Byers, Grimes; Dyer (Roberts 77), Celina (Fulton 83), Routledge (Surridge 87); Ayew. Substitutes: Nordfeldt, Cabango, Peterson, Borja.
Referee: David Webb (Lancashire)