Winger Nathan Dyer provided the cutting edge as Swansea held off Barnsley to reach the fourth round of the FA Cup.
He set up Angel Rangel for Swansea's first goal and added the third himself after good work from Wayne Routledge.
Barnsley had led after a swift counter finished by Ricardo Vaz Te, who later poked home an inswinging free-kick.
Rangel's equaliser was followed by a sublime long-range Danny Graham shot into the top corner, with the striker adding a late fourth from six yards.
Swansea boss Brendan Rodgers had resisted the urge to make too many changes, but German goalkeeper Gerhard Tremmel was given his debut in place of Michel Vorm.
Gary Monk also returned in defence for the on-loan Stephen Caulker, as parent club Tottenham did not want the youngster cup-tied.
Barnsley were able to field their on-loan star Danny Drinkwater, recovered from a thigh strain, and the Manchester United midfielder produced the Tykes' first threat on goal as he drilled a shot just over the bar from a tight angle.
Swansea should probably have been ahead before that and Barnsley were indebted to goalkeeper Luke Steele for two good early saves.
First Steele beat away a fierce 30-yard drive from Kemy Agustien, then was quickly off his line to smother the danger in a one-on-one against Routledge.
But against the run of play, Barnsley took the lead when Routledge lost possession in the Tykes' 18-yard box.
The home side launched a flowing counter-attack up the right and Craig Davies played in Vaz Te on the edge of the area, and the Portuguese struck a rising shot into the top corner of the Swansea net.
Swansea responded in style to level within a minute, as Dyer drove the visitors upfield before teeing up Rangel to slide the ball past Steele.
It took just one minute of the second half for Swansea to break Barnsley's resistance a second time, as Graham received the ball midway in the Yorkshire side's half before turning and unleashing a brilliant 30-yard shot that gave Steele no chance.
Swansea's wing play continued to torment Barnsley, as it had at Aston Villa last week, and the Welsh side's two wing dynamos combined to double their lead.
Routledge broke down the left, beating Stephen Foster before cutting the ball back for the arriving Dyer to blast gleefully home, making amends for ballooning over the bar from a good position moments before.
Swansea's grip on the game continued to strengthen but Barnsley were by no means out of it, with Vaz Te having a couple of half-chances blocked.
The former Bolton Wanderers striker had looked the most likely to double his and his side's goal tally, so it was no surprise when he was the one to get the final touch when Swansea failed to deal with Drinkwater's inswinging free-kick.
Barnsley still had 25 minutes to force a replay and now had their tails up, with Reuben Noble-Lazarus sending Tremmel scrambling across his goal as his shot veered just off target.
Scott Sinclair, on for Dyer, threatened to put the result beyond doubt but Steele did well to reach his shot, although a late off-side flag made the save academic.
With a minute of normal time remaining, Graham finally settled Swansea nerves as the striker added a fourth goal from close range on the left following good work from Routledge and Joe Allen.
Barnsley manager Keith Hill told BBC Radio Sheffield:
"We've played against superb and educated opposition. There's a patience to the way that they play, but we had to make sure we locked the doors. Once you open the game up against that type of opposition they just pick you to pieces.
"In the second half we had to give it a go but we conceded the goals at the wrong time. We're making mistakes that are contributing to us losing that game.
"We allowed the opposition to have the ball in negative areas of the pitch. The timing of their second goal really hurts and hinders our opportunity to get anything out of the game, because we know how good they are with the ball."
Swansea assistant manager Colin Pascoe told BBC Radio Wales:
"The conditions were very difficult, but both teams tried to get the ball down and pass and Barnsley deserved to take the lead. It was good from ourselves to respond quite quickly and make it 1-1.
"I wouldn't say we were in control for long periods - Barnsley were superb, with the way they passed the ball.
"We got the goals at the right time and in football if you do that it just eases a bit of pressure. Fair play to Barnsley, because they put us under a lot of pressure."