Bafe Gomis scored his first goal for the Swans to fire them into the Capital One Cup third round following a 1-0 victory over Rotherham United at the Liberty.
The French striker's neat turn and shot after 22 minutes proved the difference for Garry Monk's side, with the 29-year-old marking his first start with a goal.
The Swans enjoyed the vast majority of possession and chances, with Dwight Tiendalli and Jefferson Montero firing into the sidenetting.
Monk, who handed Federico Fernandez and Tottenham loanee Tom Carroll their debuts along with a first appearance for youngster Josh Sheehan, now sees his side move into the third-round draw.
Garry Monk made ten changes to the side that beat Burnley on Saturday.
Nathan Dyer was the only player to start both games, and the winger captained the home side in place of Ashley Williams, who was rested.
New signings Federico Fernandez and Tom Carroll made their debuts for the home side, while fellow summer recruits Jefferson Montero and Bafe Gomis made their first start for the Swans.
Youngster Josh Sheehan, who impressed during the club's summer tour of America, earned his first appearance in a midfield also containing the fit-again Jay Fulton.
Meanwhile, the visitors named former Swans loanee Febian Brandy, who featured in the club's League One title-winning campaign, on their substitutes bench.
SWANS: Gerhard Tremmel, Dwight Tiendalli, Kyle Bartley, Federico Fernandez, Jazz Richards, Jay Fulton (Jonjo Shelvey, ht), Josh Sheehan (Gylfi Sigurdsson,, 60), Tom Carroll, Nathan Dyer (capt), Jefferson Montero, Bafetimbi Gomis (Wilfried Bony, 74).
SUBS: Lukasz Fabianksi, Neil Taylor, Jordi Amat, Ki Sung-Yueng.
ROTHERHAM: Scott Loach, Kari Arnason, Kirk Broadfoot (Richard Brindley, 53), Richard Wood, Lee Frecklington, Alex Revell, Jordan Bowery, Ben Pringle, Mat Sadler (Joe Skarz, 71), Craig Morgan (capt), Michael Tidser (Conor Newton, 59).
SUBS: Adam Collin, Febian Brandy, John Swift, Matt Derbyshire.
The Swans made the early threats on goal, with captain Nathan Dyer seeing his effort blocked following neat link-up play with Dwight Tiendalli.
Jefferson Montero then laid used his pace down the left as he outwitted his marker Kirk Broadfoot just inside the Rotherham half before the Ecuadorian raced towards goal before a fine one-two with Josh Sheehan allowed the winger to again beat his marker only for his cross to be snuffed out.
Sheehan then saw his effort from inside the box deflected out for a corner, before the Swans found the breakthrough on 22 minutes.
After Dyer slipped his pass inside to Bafe Gomis, and the French striker turned his marker before shooting left-footed back into the bottom right-hand corner.
Rotherham were then dealt a decent opportunity to level as a cross was cleared straight to Lee Frecklington, but the midfielder's half-volley flew over the crossbar from 18 yards.
The Swans were soon back on the attack, and when Jay Fulton's chipped pass sent Gomis down the left, the striker made inroads towards the box before squaring to Montero, but his curling effort was comfortably dealt with by keeper Scott Loach.
New signing Federico Fernandez then headed over from an inswinging Sheehan free-kick before Tiendalli fired into the sidenetting after Gomis, Sheehan and Dyer linked up.
HALF-TIME: Swans 1 Rotherham 0
Garry Monk made a change at the break as Jonjo Shelvey came on for Fulton, but it was Rotherham who created the first big chance.
Kari Arnason's sublime through ball sent Alex Revell in on goal, but a perfectly-timed sliding challenge from Kyle Bartley blocked the striker's effort some ten yards out.
Rotherham pushed again, and Jordan Bowery did well to hold off Tiendalli before his low, curling effort bounced off the surface before Gerhard Tremmel gathered.
Then Shelvey tested Loach from long range, with his 25-yard effort being palmed away at full stretch by the visiting keeper.
Monk made another change on the hour, with Gylfi Sigurdsson coming on for Sheehan, while, later on, Wilfried Bony replaced Gomis, who received a standing ovation from the Jack Army.
The Swans continued their search for a second goal, and Montero went close as he fired into the sidenetting following a quickly-taken corner in the last 15 minutes.
Then Sigurdsson failed to make the most of a deep cross from Jazz Richards after the full-back had raced down the left to get onto the end of Montero's through ball.
Montero was causing regular problems to the Millers defence with his electric pace, and he again created an opportunity as he crossed for Bony, but the Ivorian's effort was blocked.
Bony again tried his luck, this time after Dyer's cross, but the substitute could only direct his first-time effort over the crossbar from 12 yards.
But Rotherham kept plugging away, and the visitors were handed a fine opportunity to level as sub Conor Newton found himself in space some 15 yards out, but his shot whistled past the post.
Then, late on, Bony saw his rasping drive fly past the post in what proved the final action of the tie.