United's 2018/19 campaign began with an unfortunate defeat at Bramall Lane in front of the Sky cameras.
George Baldock opened the scoring just after the hour, but goals from Oli McBurnie and Yan Dhanda saw Swansea come from behind to snatch all three points late on.
For the televised opener, Wilder named three of his new signings in the starting XI. Henderson, Egan and McGoldrick all made their competitive home debuts, whilst Woodburn was on the bench.
The Blades received a raucous welcome to the pitch, but it was Swansea who tested Henderson twice in the opening exchanges from two free-kicks. United hit back though and after Clarke had out run Van Der Hoorn and helped win a corner, Evans played it short to McGoldrick before whipping in a cross from the left that narrowly evaded the run of O'Connell and Basham.
The game ebbed and flowed, with no goalscoring chances of note as the clock ticked past the half hour mark, although tempers frayed somewhat when Stevens caught Fernandez accidently, prompting a reaction from City's bench.
Moments later, United had appeals for a penalty waved away by referee Simpson as Basham went down under a challenge from Fernandez. Fulton then lost the ball on halfway and McGoldrick saw his effort deflected over for a corner, from which O'Connell headed over at the back post, the last meaningful action of the half.
The Blades were almost caught out at the beginning of the second period after McKay got the better of Basham, but a combination of O'Connell and Egan blocked Asoro's goal bound effort. At the other end, United had the best chance of the game in the 58th minute, Celina carless gave away the ball and McGoldrick advanced towards goal before dragging his effort wide.
A minute later, United were gifted an even better opening, McGoldrick teed up Lundstram, but the midfielder sliced his effort wide from an excellent position, although he looked to make amends immediately chasing 50 yards before tackling Olsson. In response, Carroll fired an effort over.
But the Blades did finally take a deserved lead just after the hour, some neat play in the left resulted in Lundstram cutting the ball back for BALDOCK to side foot past the desperate dive of Nordfelt.
Swansea threatened a response five minutes later, but Henderson was at his best to tip over Fulton's looping header over. But the equaliser came in their next attack, a swift move forward, saw Fleck and Henderson try to avert the danger, but the ball fell kindly MCBURNIE who fired into the net.
City lifted by restoring parity sensed a second, and it almost came from the head of McBurnie after Montero had got the better of Baldock. And they came even closer in the 83rd minute as McKay rattled the crossbar, from another Montero centre.
With four minutes remaining, Swansea did take the lead. Montero again got free on the right and McKay teed up substitute DHANDA to score with his first touch. United tried to respond and almost got a helping hand from Roberts, but Nordfelt tipped the ball wide for a corner, from which Evans headed over.
Four minutes of added time we signalled, but despite United's best efforts, City held on to claim the win, with the last action seeing Egan's effort saved by Nordfelt.
Blades: Henderson, Baldock, Stevens, Fleck, O'Connell, Basham (Duffy 87), Lundstram (Sharp 79), Evans, Clarke, Egan, McGoldrick (Woodburn 72).
Unused: Leonard, Stearman, Lafferty, Moore.
Swansea: Nordfelt, Olsson (Grimes 72), Van Der Hoorn, Fernandez, McBurnie, Celina, Carooll, Asoro (Montero 63), McKay, Roberts, Fulton (Dhanda 85).
Unused: Dyer, Roden, Amat, Mulder.