Record signing Pablo Hernandez grabbed his first goal for the club in the 65th minute and Michu's sixth of the campaign quickly doubled their advantage.
Emmerson Boyce pulled one back almost immediately for Wigan, but Arouna Kone had an equaliser ruled out for offside as their own winless run extended to six games.
The pre-match focus was on the fact that neither side wore the anti-racism Kick It Out t-shirts during the warm-ups, understood to be a gesture of solidarity from the respective squads towards players who did not wish to wear them.
But it could not hide how important a game this was for both sides, particularly for the hosts, who had been stung by reports of dressing-room unrest and dissatisfaction with manager Michael Laudrup's methods.
Swansea knew a win would go some way to silencing such talk and, with top scorer Michu up front instead of Danny Graham, they started purposefully.
It was the visitors, though, who had the best early chances, Kone and Shaun Maloney both being denied by sharp saves from Michel Vorm.
Michu had a great chance at the other end but could not keep his volley down from Wayne Routledge's deft pass but the game was proving a scrappy affair indicative of both sides' recent struggles.
Michu headed over the bar after some slick build-up involving Ki Sung-yueng, Leon Britton and Wayne Routledge on the half-hour and it needed fine clearances from Jean Beausejour and Ivan Ramis when there was more danger down the Wigan left seconds later.
Swansea were almost caught on the break when Maloney evaded Angel Rangel and played Beausejour in, but the Chile international could only fire over Vorm's bar.
Routledge was at the heart of Swansea's better moments and his superb flick and volleyed pass with two minutes left in the first half set Michu free to play a ball across the box, which saw Beausejour block a Hernandez effort.
Swansea pressed again on the resumption and Britton whistled a shot beyond Ali Al Habsi's despairing dive, but also past the left-hand post.
They were soon on the front foot again and it took a strong parry from Al Habsi to beat out Ki's strike from Routledge's neat reverse pass.
The breakthrough finally came after 65 minutes. Routledge brilliantly beat Ramis out on the left and when Jonathan de Guzman fed Hernandez, the Spaniard swivelled and dispatched a right-foot shot low into the net.
It was 2-0 barely two minutes later. De Guzman's superb inswinging corner picked out Michu's near-post run for the forward to glance home his sixth league goal of the campaign.
The unexpected glut of goals continued as Wigan instantly pulled one back. Chico Flores misjudged a long ball, and his error ended with Boyce acrobatically flicking in from James McCarthy's scuffed shot.
Kone thought he had equalised during the chaotic spell, but his far-post header was chalked off for a tight offside decision.
Swansea, who were by now guilty of dropping ever deeper in defence, were denied a third as Al Habsi pushed away a rising strike from substitute Nathan Dyer after an excellent Swansea break.
Franco Di Santo then wasted Wigan's best opportunity as he fired tamely at Vorm when the ball broke to him in the box, while the Dutchman had to save from opposite number Al Habsi as Swansea held on in a frantic finish.