Yan Dhanda scored with his first touch in senior football as the Swans came from behind to give boss Graham Potter a winning start to his Liberty reign.
The Swans had to come from behind at Bramall Lane after George Baldock’s 62nd-minute goal gave Sheffield United the lead.
Oli McBurnie ensured the Swans were back level within nine minutes as he fired home his first league goal for the club.
Then, after only being on the pitch for 29 seconds, Dhanda popped up with a thrilling late winner as he netted from close range.
Potter, who joined the club in June following a hugely successful spell in Sweden with Ostersund, handed new signings Bersant Celina, Joel Asoro and Barrie McKay their debuts.
Kristoffer Nordfeldt started in goal while Martin Olsson got the nod at left-back after proving his fitness following his return from World Cup duty with Sweden.
In midfield, Jay Fulton – who spent the second half of last season on loan at Wigan - made his first Swans appearance since the 1-1 home draw with Crystal Palace in December.
He was joined by Tom Carroll and Celina, who arrived on a permanent deal from Manchester City this week.
There were new faces out wide as Asoro and McKay, who arrived from Sunderland and Nottingham Forest respectively, were given an immediate chance to impress.
It proved a low-key affair in the first half, with few chances created by either side.
The game livened up after the break, with Asoro seeing his effort blocked after a fine interception and run by the lively McKay.
At the other end, a misplaced pass from Celina found McGoldrick, but his former Ipswich team-mate failed to hit the target.
A minute later, Connor Roberts’ throw was intercepted by John Lundstrom, and again he fired wide.
Baldock wouldn’t miss three minutes later though, as he finished coolly from ten yards.
Jay Fulton then saw his flicked header tipped over the crossbar before McBurnie levelled after a sweet Swans move.
McKay then saw a thunderous effort rattle off the crossbar, but the Swans' chance of victory was not gone.
With four minutes left, sub Montero’s cross found its way to Dhanda, and he steered the ball into the net with his first touch.
It was the perfect debut for Dhanda, a perfect result for the Swans and a perfect start for Potter.
Sheffield United: Dean Henderson, Chris Basham (Mark Duffy, 87), John Egan, Jack O’Connell, Enda Stevens, George Baldock, John Lundstram (Billy Sharp, 79), Lee Evans, John Fleck, David McGoldrick (Ben Woodburn, 72), Leon Clarke (capt).
Subs: Simon Moore, Ryan Leonard, Richard Stearman, Dan Lafferty.
Swans: Kristoffer Nordfeldt, Connor Roberts, Mike van der Hoorn (capt), Federico Fernandez, Martin Olsson (Matt Grimes, 72), Jay Fulton (Yan Dhanda, 85), Tom Carroll, Bersant Celina, Joel Asoro (Jefferson Montero, 64), Barrie McKay, Oli McBurnie.
Subs: Erwin Mulder, Joe Rodon, Jordi Amat, Nathan Dyer.