After a drab first half, the Blades took the lead via George Baldock.
But the introduction of Jefferson Montero turned the game in Swansea's favour, with goals from Oli McBurnie and Yan Dhanda.
The Blades gave club record signing John Egan his debut, alongside goalkeeper Dean Henderson and striker David McGoldrick.
Club captain Billy Sharp, Mark Duffy and Liverpool's on-loan Ben Woodburn were named on the bench.
United enjoyed a bright start, but it was the visitors who had the best chance in the opening 10 minutes.
Martin Olsson fired in a 25-yard free-kick, but Henderson - on loan from Manchester United - was up to the task.
At the other end, Leon Clarke raced away down the left flank, crossed, but McGoldrick was unable to bring the ball under control and the chance was gone.
Chances were at a premium in the opening half, and when Lee Evans found himself on the edge of the penalty box with a shooting chance - after Clarke's lay-off - the Blades midfielder would have been disappointed to see his effort charged down by the Swans defence.
In front of a 24,654 crowd, the game was starting to get stretched, and United twice had good chances to break the deadlock.
First, McGoldrick fired wide, then moments later, John Lundstram also failed to hit the target as he raced into the box.
So it was no surprise when United took the lead 0n 62 minutes, Baldock popping up in the box to slot the ball home.
Henderson had to be alert to tip over Jay Fulton's looping header as Swansea searched for an equaliser.
It came moments later. A slick build-up from the visitors saw Henderson deny Connor Roberts, before McBurnie - the former Barnsley loan striker - hammered in the rebound.
And it could have been worse for United, Montero's cross finding McBurnie, but his deft header just faded beyond Henderson's left-hand post.
Then, from another teasing Montero cross, Barrie McKay smashed a shot against the crossbar.
Swansea were on top and it was no surprise when they edged in front. The ball was moved the length of the field - again Montero was key - and newly-introduced substitute Dhando swept the ball home with his first touch.
Swans goalkeeper Kristoffer Nordfeldt had to be alert to prevent a Roberts own goal, while substitute Sharp scuffed his late chance to snatch an equaliser.
Blades boss Chris Wilder said: “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.
“We’re still light and we’re still looking. The search goes on.”