Two goals from Danny Graham helped the Swans march into the fourth round of the FA Cup after a hard fought win over Barnsley at Oakwell.
The home side took the lead midway through the first half through Ricardo Vaz Te but Angel Rangel levelled a minute later with a cool finish.
Danny Graham then netted his first straight after half time with a rasping 25-yard drive before Nathan Dyer slotted home Wayne Routledge's cross.
Vaz Te scored his second to set up a tense finish but Graham added a fourth in the last minute as the Swans booked their place in the next round.
Brendan Rodgers made two changes to the side that recorded a first away win of the season at Aston Villa on Monday.
German keeper Gerhard Tremmel made his competitive debut for the Swans in place of Michel Vorm, while captain Garry Monk replaced Steven Caulker after the defender was denied permission to play by parent club Tottenham.
New signing Gylfi Sigurdsson&nbsp;was named in a strong substitutes bench that included Joe Allen, Leroy Lita and Scott Sinclair.
Barnsley: Steele, Hassell (WIseman, 58), McEveley, Foster, Davies, Done (Noble-Lazarus), Perkins (Gray, 58), McNulty, Addison, Vaz Te, Drinkwater
Subs not used: Preece, Edwards, Taylor, Digby
Swans: Gerhard Tremmel, Angel Rangel, Garry Monk (capt), Ashley Williams, Neil Taylor, Leon Britton, Kemy Agustien (Joe Allen, 63), Andrea Orlandi (Gylfi Sigurdsson, 60), Wayne Routledge, Nathan Dyer (Scott Sinclair, 77), Danny Graham
Subs not used: Jose Moreira, Fede Bessone,&nbsp;Leroy Lita, Luke Moore
Referee: Stuart Attwell
Attendance: 7,380 (including 1313 away)
After a scrappy opening ten minutes in which both sides tried to get a foothold in the game it was the Swans who had the first effort on goal as Kemy Agustien unleashed a powerful drive from 25 yards that Luke Steele did well to parry away.
Just a minute later and the visitors had a golden chance to open the scoring when Wayne Routledge latched onto Andrea Orlandi's wonderful pass and as the winger beared down on goal, Steele was equal to it once more with a fine save with his legs.
As the Swans began to assert their dominance on the game, particularly in midfield, Barnsley had their best chance of the opening half hour when on-loan Danny Drinkwater drilled just over the bar from 25 yards out.
However, despite the visitors having the lion's share of the game, the home side took the lead after half an hour when Ricardo Vaz Te latched onto Neil Taylor's sliding tackle to drive at the Swans defence before firing a shot that deflected off Garry Monk and into the top corner.
But the home crowd barely had time to sit down before Swansea drew level less than a minute later. After left back Jay McEveley was disposessed by Dyer, Orlandi threaded a lovely ball into the path of Angel Rangel who took a touch before sliding the ball past the onrushing Steele.
And Dyer himself had a chance to put the Swans ahead shortly after as he cut in from the right corner of the box before firing a curling effort that just cleared the crossbar.
The winger was now causing McEveley all sorts of problems down the right as he beat the full back before clipping a teasing cross into the box for Danny Graham, but Steele did well to palm the ball out for a corner.
Half time: Barnsley 1 Swans 1
The Swans made a perfect start to the second half as they went in front within a minute of the restart.
Kemy Agustien played the ball into Danny Graham's feet and as the striker turned away from his marker, he fired a powerful 25 yard shot that arrowed into the top corner giving Steele no chance.
Just five minutes later and Dyer had a great chance to extend the Swans' lead when he connected with Agustien's centre, but the winger scooped his effort high over the bar.
However, the 24-year-old didn't have to wait long to score his second goal in as many games as Routledge beat Foster before cutting the ball back for Dyer who turned the ball past Steele from six yards.
Shortly after Vaz Te had a couple of efforts blocked for the home side, the travelling support had their first chance to see new signing Gylfi Sigurdsson as the Iceland international was introduced on the hour mark in place of Orlandi.
But the home side refused to give in and pulled a goal back&nbsp;on 65 minutes&nbsp;when Drinkwater's right wing free kick wasn't dealt with by the Swans defence and Vaz Te poked home at the far post for his second of the game.
And they could have levelled shortly after when substitute Reuben Noble-Lazarus connected with Vaz Te's deflected cross, but the youngster screwed his volley well wide.
Despite a period of pressure from the home side, the Swans dominated the final ten minutes and sealed their place in the next round when Graham turned home Routledge's cross in stoppage time.