The Swans still occupy sixth spot following this comfortable home win against mid-table Barnsley.
Goals before the half-hour from Ashley Williams and Shefki Kuqi put the Swans in the driving seat only for Darren Moore's header to halve the deficit before the break.
The visitors did have their chances to claw it back to 2-2 but a fine Darren Pratley finish shortly before the end ensured that maximum points would remain in south west Wales.
Fede Bessone made his long awaited return to start against Barnsley at the Liberty.
The Argentinian full-back, who came in for David Edgar, made his first start since limping out of the home fixture against Newcastle two months earlier with a knee injury.
His inclusion was one of six changes by the home side - Albert Serran started in the league for the first time this year as a replacement for the injured Angel Rangel while Leon Britton, Mark Gower, Cedric van der Gun and Shefki Kuqi were also included.
Swans:Dorus de Vries; Albert Serran, Garry Monk(C), Ashley Williams, Fede Bessone; Leon Britton, Mark Gower (Andrea Orlandi, 70'), Darren Pratley, Cedric van der Gun, David Cotterill (Tom Butler, 89'); Shefki Kuqi (Lee Trundle, 77').
Not Used:David Cornell (GK), Gorka Pintado, Joe Allen, David Edgar.
Barnsley: Lukle Steele, Luke Potter, Darren Moore, Stephen Foster (C), Ryan Shotton, Adam Hammill, Hugo Colace, Nathan Doyle (Bobby Hassell, 69'), Iain Hume (Jamil Adam, 53'), Martin Devaney (Reuben Noble-Lazarus, 81'), Andy Gray.
Not Used: David Preece (GK), Jon Macken, Jacob Butterfield, O'Neil Thompson.
Referee:D Phillips (Sussex)
The glorious sunshine was beating down on the Liberty Stadium as the home side upped the tempo from the first blow of the referee's whistle.
Dubbed the first of three of the club's biggest matches in recent history, victory against the Tykes was needed to ensure that the Swans' spot in the play-off's would strengthen.
Returning full-back Fede Bessone was able to win his side an early corner while a couple of weak attempts were easily collected by Luke Steele.
But the Tykes' stopper was guilty of dropping a huge clanger with four minutes played - what looked like to be an easy collection from Garry Monk's long clearance forward, Steele totally missed the ball which fell kindly for Shefki Kuqi with an open goal gaping in front of him.
All that was required of the Finnish international was to tap home the opener but, somehow, on his weaker left foot, he was unable to find the target.
But that didn't affect the attitude of players as they continued to battle in their bid to score the game's first goal, but Barnsley were showing that they were not going to be pushovers as they looked dangerous through a couple of counter attack moved.
Darren Pratley forced Steele into saving his 20th minute strike but, three minutes later, the Swans got their reward.
Supporters were certainly becoming accustomed to seeing their side find the net from set-pieces and they were rewarded with another on this occasion - Cotterill's corner was initially met by the head of Cedric van der Gun, his attempt was saved but Ashley Williams was ideally placed to fire the second ball past the goalkeeper.
The Swans were well in control and after Adam Hammill had a penalty appeal turned down for the visitors the action switched back to the other end of the pitch where the busy Steele was alert to deny David Cotterill's curler.
And the game wasn't even half-hour old when Kuqi was able to make ammends for his earlier miss to score his first goal at the Liberty - Mark Gower's well-timed delivery into the heart of the area hit the right spot for the Finn to fling himself towards it and glance a header into the bottom corner.
But that two-goal advantage lasted just five minutes as the visitors struck with a set-piece of their own; Dorus de Vries came out to collect Iain Hume's corner but big defender Darren Moore beat him to it to power a header through a crowd of players and into the net.
The Swans responded well with Cotterill heading just over the bar and Kuqi, two minutes before the break, seeing a shot from the angle pushed away by Steele.
But it could easily have been 2-2 going into the break had it not been for Dutchman de Vries as he denied Hugo Colace from point blank range following a scramble which the defence failed to clear.
Half-Time: Swans 2 Barnsley 1
Iain Hume screwed an excellent chance wide of goal within 10 seconds of the re-start and had it gone in would have been harsh on the home side who had controlled the vast majority of the first-half.
The Canadian international was replaced shortly afterwards which was just before David Cotterill shot low into the hands of Steele following a move set-up by van der Gun, and then just before the hour the Dutchman won his side a corner when his effort was pushed around the post.
But the visitors certainly were not down and out as Colace unleashed a simmering over the top of goal while Martin Devaney failed to connect properly with his chance inside the area.
Andrea Orlandi was introduced as the Swans' first sub of the afternoon with 20 minutes left on the clock and within minutes of his arrival the midfielder set-up the hosts' third.
He rode the challenges of two defenders as he galloped towards the byline, looked up before crossing low to top-scorer Darren Pratley just inside the area and, with his first touch, curled right-footed into corner of the net.
Trickster Lee Trundle was next on the scene and within minutes of his arrival the striker's decision to take on the entire Barnsley defence could have paid off had his low effort, saved by Steele, had more behind it.
It was just a case of keeping a hold of the ball in the closing stages before the referee brought this contest to the end to keep the Swans in the hunt for a top-six final finish.