The Wearsiders had their captain Lee Cattermole - a former Latics player - sent off in the first half of Saturday's Premier League encounter but took the lead in the 66th minute when substitute Gyan converted Jordan Henderson's cross.
The hosts came away with a point though thanks to Alcaraz, whose effort three minutes from time caught Black Cats goalkeeper Simon Mignolet cold.
Wigan, who started the day without a goal or a point at home after being hammered 4-0 by Blackpool and 6-0 by Chelsea, created the first opening in the eighth minute when Mauro Boselli headed the ball into the path of Mohamed Diame, who lashed it across Mignolet and wide.
Two minutes later Hugo Rodallega cut into the Black Cats box but saw his shot deflect off Nedum Onuoha and behind for a corner before Hendry Thomas received the first caution of the afternoon from referee Andre Marriner.
Cattermole, making his first return to the DW Stadium since leaving Wigan for Sunderland in 2009, then picked up a slightly harsh yellow card for a challenge on Diame, to the delight of the home supporters.
Emmerson Boyce followed him into the referee's book moments later, but it was Cattermole again in trouble within seconds, scything down Rodallega on the edge of the box and he was given his marching orders.
It was Cattermole's second sending off in three league games following his dismissal against Birmingham on the opening day of the season and his third since joining Sunderland.
Once the midfielder had departed the pitch with jeers ringing around the ground, Rodallega hit the resulting free-kick onto the roof of Mignolet's net.
The game was becoming increasingly ill-tempered and after Mignolet had punched a cross away from danger with Boselli lurking, Steve Gohouri was next to be booked for a rash challenge on Darren Bent.
Ahmed Elmohamady then had to be substituted after appearing to take a knock, with Cristian Riveros sent on as his replacement.
Wigan looked to make the most of their man advantage and James McCarthy saw his effort go narrowly over, but referee Marriner was soon called into action once more, this time showing Danny Welbeck yellow for fouling Ronnie Stam.
Sunderland boss Steve Bruce brought on the club's record signing Gyan for Welbeck at half-time in an attempt to rejuvenate his troops but it was Wigan who remained on the front foot, with Rodallega firing at Mignolet after collecting the ball on the edge of the box and sending another effort just wide when he might have been expected to score.
Wigan manager Roberto Martinez then introduced one of his new signings, Franco Di Santo, as well as Charles N'Zogbia with Thomas and Stam making way.
Boselli managed to squirm around Mignolet after being fed by Diame, but could not get a shot away and Di Santo delivered a cross into the danger zone which the Belgian stopper dived to catch.
Boselli then had another attempt which went high over the bar, but Latics fans were sensing that a goal was not far away.
That proved to be the case, but it was Sunderland who found it against the run of play as Gyan volleyed Henderson's inch-perfect cross past Ali Al Habsi.
Gyan looked to double his tally and the visitors' lead moments later with an attempt that skewed well wide.
Boselli then had a glorious chance to equalise but contrived to send his header from Di Santo's cross wide with the goal gaping.
The striker was almost immediately taken off, with Martinez introducing Tom Cleverley in a bid to save the game and the on-loan Manchester United youngster made a telling contribution as Wigan drew level in the 87th minute.
Mignolet attempted to clear with a punch but Cleverley sent the ball back towards goal with a fierce drive.
It was met by Latics skipper Alcaraz, whose header Mignolet was unable to prevent from slipping over the line.