Victor Moses put the struggling Latics ahead on the stroke of half-time, but Gary Caldwell was then sent off for deliberate handball in the 76th minute and Jonathan Walters converted the resulting penalty to draw eighth-placed Stoke level.
When substitute Cameron Jerome - whose header had led to Caldwell's dismissal - put the hosts ahead with six minutes left, things looked bleak for Wigan, but Watson then scored a late penalty of his own - with his first touch - after Ryan Shotton had been penalised for a tug on Hugo Rodallega's shirt.
Roberto Martinez's side remain in the Barclays Premier League relegation zone, but the Spaniard will surely take heart from the fight his side showed in salvaging something from the game.
The visitors had slipped a place to 19th by kick-off following Blackburn's sensational 3-2 victory at Old Trafford, and they were forced onto the back foot early on.
Jordi Gomez and Rodallega each warmed the palms of home goalkeeper Thomas Sorensen as Wigan settled, yet Stoke remained on top and Robert Huth headed narrowly wide from a Rory Delap throw-in.
Latics stopper Ali Al Habsi was soon inconvenienced by another Delap missile, before the veteran midfielder fired narrowly wide from 20 yards.
At the other end, Ronnie Stam was given the opportunity to shoot from distance and Sorensen had to be alert to tip away his dipping effort.
After the fit-again Andy Wilkinson had gone close for Stoke in an increasingly open game, Rodallega then latched onto a long ball and found himself through on goal, but Ryan Shawcross intervened with a crucial clearing header after the striker had lifted the ball over the onrushing Sorensen.
Huth then headed over from another set-piece and Marc Wilson tried his luck for the hosts with a long-range effort that sailed wide.
If that was ambitious, then Maynor Figueroa's subsequent shot from inside his own half represented the most optimistic of attempts, although the same player had scored from a similar distance at this ground in December 2009.
Stam prompted fury among the Stoke support - and earned himself a booking - shortly before the interval when he blasted the ball into the crowd after it had gone out of play.
Wigan then moved ahead in the 45th minute as Stam played in Rodallega down the right channel and the Colombian crossed for Moses to slot home with a clinical first-time finish into the roof of the net.
Stoke began the second half with renewed vigour, but were forced into a change in the 49th minute as Wilkinson limped off to be replaced by Shotton.
As had been the case in the opening period, the Latics weathered the early pressure and gradually settled into their stride.
Yet they survived a major let-off just before the hour mark when another long throw from Delap picked out Huth, whose firm header struck a post before bouncing back into the arms of a grateful Al Habsi.
Although Wigan were content to play largely on the counter, they retained an attacking threat and the impressive Mohamed Diame saw a 30-yard strike deflect narrowly off target before Rodallega nodded wide from a corner.
As the home supporters grew increasingly frustrated, Tony Pulis opted to introduce Jerome in place of Matthew Etherington with 17 minutes remaining and the move reaped immediate rewards.
After Peter Crouch had diverted a deep cross from the right back across the area, Jerome's goalbound header was blocked by the arm of Wigan skipper Caldwell, who was duly sent off.
Walters drilled the resulting penalty into the bottom-left corner and Stoke were soon pouring forward in search of a second.
Jerome was the man to provide it, the former Birmingham man slotting a cool finish into the far corner after flick-ons from Crouch and Walters had sent him clear.
However, Wigan were then awarded a penalty of their own after Shotton grabbed Rodallega's shirt in the box.
Watson climbed off the bench to take the spot-kick and the midfielder calmly sent Sorensen the wrong way with his first touch of the game.
Huth saw another header fly wide as the final whistle approached, while appeals for a second Stoke penalty were waved away after Shawcross took a tumble in stoppage-time.