The Blades were unable to cling on after George Baldock’s opener, relegated Swans earning three points with goals from Oli McBurnie and Yan Dhanda.
A combination of missed chances – debutant David McGoldrick and John Lundstram spurned chances to score – and sloppy defending meant the Blades, who travel to Middlesbrough tomorrow night, emerged empty-handed.
It left Wilder, who has a club record £5m offer for Ipswich Town striker Martyn Waghorn on the table, eager to bring in a forward before Thursday’s transfer window closes.
Despite facing a team who were plying their trade in the top flight last season, Wilder said: “A loss is a loss whether at the start or end of the season and it still hurts. It should hurt the players, it hurts myself, and it hurts our supporters.
“We don’t do unlucky, we have not defended properly. It’s not unlucky, it’s just poor defending, and not putting the ball in the back of the net when we had two really good opportunities.
“We have worked hard. When we have opportunities to score we have to take them.
“This is the Championship, you don’t get seven or eight golden chances like we did in the division below.
“We are just a bit disappointed we didn’t take them. We had a good period after half-time when we were on the front foot. David and John had opportunities and then scored a great goal through George.
“But credit the opposition, they came back into it, they weren’t just going to lie down and let us walk over them. We have been punished for not defending properly.”
The opening 45 minutes were something of an anti-climax for the 24,654 supporters inside Bramall Lane after a long, hot summer.
Neither side could muster a clear scoring opportunity with Blades goalkeeper Dean Henderson – making his debut alongside McGoldrick and John Egan – having little involvement.
But United started to build a head of steam in the second half as they kicked towards the Kop.
A misplaced pass gifted the ball to McGoldrick, but with time to pick his spot the former Ipswich striker dragged his shot wide.
Moments later Lundstram charged into the penalty area, but failed to hit the target.
When United’s goal finally came it was from the unlikely source of wing-back Baldock.
The 25-year-old netted just once last season, but matched that tally with the new campaign just 62 minutes old.
Good play down the left flank ended when Enda Stevens’s cross picked out Baldock, who side-footed the ball beyond Kristoffer Nordfeldt.
The goal seemed to spark the visitors into life, Henderson – on loan from Manchester United – doing well to tip over Jay Fulton’s looping header.
But the Blades goalkeeper had little chance with either of Swansea’s goals, both classic counter-attacks.
The first came nine minutes after going behind. The impressive Barrie McKay, on debut, played a one-two with McBurnie before racing into the area.
Henderson did well to save the initial effort – which looked to come off retreating Blades striker Leon Clarke – but McBurnie pounced to equalise.
The 22-year-old striker had a profitable loan spell at Barnsley in the second half of last season – netting nine times, including a goal against the Blades – and looks a real prospect.
The introduction of substitute Jefferson Montero on the left wing – with McKay on the right – caused United a real headache.
McBurnie’s header, from a Montero cross, skipped wide before the winger teed up McKay, who smashed his volley against the crossbar.
It was the pace and trickery of Ecuador international Montero that created Swansea’s winner. He attacked the retreating Blades defence and his cross was poked home by Dhanda – the substitute netting with his first touch in senior football.
The closest the Blades came to an equaliser saw Connor Roberts divert a cross towards his own goal, forcing Nordfeldt into a lunging save.
Wilder said: “You can’t defend like that against quality players. You’ve got to defend properly.
“It was a tight game and we started off okay. To go one-nil up on day one was a good position, but the two goals were very disappointing from our point of view.
“We went to press in the corner and we didn’t need to. The second goal was a really poor one from our point of view. All of a sudden we’re back-tracking. I think we had the best opportunities so they are fine dividing lines.
“The worst it should have been was a draw. A Swansea win was the least likely of the three results. We were up against a good side. They’ve got World Cup players and really good players.
“I don’t think we should be expecting to steamroller a side that’s just come out of the Premier League.”
Sheffield United: Henderson, Baldock, Stevens, Fleck, O’Connell, Basham (Duffy 87), Lundstram (Sharp 79), Evans, Clarke, Egan, McGoldrick (Woodburn 72). Unused substitutes: Leonard, Stearman, Lafferty, Moore.
Swansea City: Nordfelt, Olsson (Grimes 72), Van Der Hoorn, Fernandez, McBurnie, Celina, Carooll, Asoro (Montero 63), McKay, Roberts, Fulton (Dhanda 85). Unused substitutes: Dyer, Roden, Amat, Mulder.
Referee: J Simpson (Lancs).