Wednesday started the better of the two, however, they poorly gifted the lead to their visitors, as a mistake from Tom Lees allowed Jamal Lowe to fire home an easy opener.
The Swans rarely got out of second gear but confirmed the win with just over a quarter of an hour to go as Jay Fulton headed home.
City are now unbeaten in four league games against the Owls with a victory that puts them within seven points of second-placed Watford.
Story of the match
Sheffield Wednesday started the game brightly, holding possession on halfway and in and around the Swansea area well, as they looked to cause an upset against the playoff-chasing visitors.
Andre Ayew had the game’s first chance in the fourth minute, as the winger wrong-footed Julian Börner before drilling the ball wide at the near post.
The Owls continued to threaten, with Barry Bannan being the instigator to Wednesday’s attacks. The captain slipped Liam Palmer down the right, but the wing-back’s pull-back was defended well by Ryan Manning, who shut down the pass towards Josh Windass.
On 12 minutes, Wednesday again were lurking on the edge of the Swans’ box, going close with a distanced effort from Joey Pelupessy, which was blocked by the feet of Fulton.
Just before the 20th-minute mark, Freddie Woodman gifted Wednesday an opportunity, as he made a hash of a dangerous cross from Windass, however, Jordan Rhodes’ header back into the middle was cleared before Palmer could take advantage.
The skewed clearance ended in City conceding a corner, from which Rhodes looped an unchallenged header over the bar.
The best chance of the first half fell to Liam Shaw, whose rifled effort from the edge was superbly palmed wide by Woodman.
However, after a positive first half-hour from the home side, Swansea were gifted the lead by an unforced error from Lees.
Bannan played a simple pass back to Lees, who opted to let the ball run across his body, which allowed Ayew to close him down and dispossess the defender. The winger raced towards goal before rolling the ball into the path of Lowe, who thumped home his 12th league goal of the season.
Ayew, Wayne Routledge and Manning all went close in quick succession as Swansea ended the half strongly.
Wednesday, who had previously lost all 16 games that they had conceded first in, knew that they had to start well to try and get at least a point out of the game to keep their dying survival hopes alive.
On 48 minutes, Lees stepped out of the defence and played a smart ball through to Windass, who drifted past several players before his left-footed effort was deflected wide by Ryan Bennett.
Moments later, a quick free-kick from Bannan released Windass down the right flank. But the 27-year-old’s cross had just too much power on it, as Shaw narrowly missed it at the back post.
Wednesday continued to press for an equaliser, with Darren Moore, who returned to the dugout after missing three games due testing positive for Covid-19, making a double substitution as Callum Paterson and Izzy Brown replaced Rhodes and Shaw on 59 minutes.
Swansea should have scored a second just after the hour mark. Börner’s poor headed clearance allowed Conor Hourihane to nod the ball into the path of Ayew, who lifted it over the head of Adam Reach before seeing his half-volley turned wide by Keiren Westwood.
With 20 minutes remaining, Bennett made an important intervention to steer a great delivery from Palmer away from the head of Paterson, who was lurking at the back post.
Connor Roberts was introduced on 73 minutes and made an instant impact as he assisted the away side’s second of the game, putting in a fine cross which was headed home by Fulton.
Swansea confidently saw out the remaining minutes to remain third in the Championship but close the gap on second place, whilst the Owls stay seven points from safety, having played a game more than 21st-placed Coventry City.
Swansea must continue to build momentum for the play-offs
The Swans’ performance was certainly not one of a promotion-chasing side, however, they ensured they got the job done without rarely getting out of second gear, defending well for the most part.
City then welcome Derby County in their final home game of the campaign before facing a Watford side, who may have already clinched an instant return to the Premier League by that point.
The win in S6 means that Swansea now have a nine-point gap between themselves and seventh-placed Reading.
This rather favourable final few fixtures will allow the Swans to cement their place in the top six and build momentum heading into the playoffs.
Man of the match
Fulton was superb for the Swans, as he marshalled the midfield well and headed home his third league goal of the season in the second period.
The 27-year-old maintained a 72% passing accuracy as he influenced the game from the middle of the park.