Cresswell hooked home his 11th league goal of the season just after the hour and then saw his late effort go in off Swans goalkeeper Dorus de Vries to earn his side three points after what had been for the most part an insipid performance.
Swansea dominated for large swathes of the game and their inability to put the ball into the back of net came back to haunt them again.
The defeat saw Paulo Sousa's men slip out of the play-off positions with one game to go and they must now beat Doncaster on the final game of the season and hope that Blackpool fail to beat Bristol City if they are to make the end-of-season showpiece.
The Blades bow out of action at Bramall Lane for the season with just one defeat in their last 15 games there, but they will consider their season to be a disappointment after a play-off bid fizzled out weeks ago.
United named Paddy Kenny in their starting line-up for the first time since last season's play-off final defeat by Burnley at Wembley following a ban for failing a drugs test.
And after a quite opening, it was Kenny that made the first meaningful contribution of the game when he got down low to keep out Darren Pratley's effort.
Swansea, desperate for the three points to nail down their spot in the top six, were beginning to look dangerous and they created several chances.
Shefki Kuqi had an effort blocked by Chris Morgan and Cedric van der Gun dragged a shot wide from 10 yards soon after.
The Swans continued in the ascendancy and felt they should have been awarded a penalty just before the break.
Pratley advanced towards the area and as he passed Nick Montgomery, he was cynically brought down on the line of the area. But referee Scott Mathieson, in discussion with his assistant Amy Fearn, decided that the infringement was outside the area.
The Swans almost made the most of the situation anyway, but Federico Bessone's curling effort produced a stunning save from Kenny.
The dangerous Pratley was a constant menace for Swansea as they upped the ante after the break and he had three chances to open the scoring in quick succession.
First his header was blocked by team-mate Kuqi, then a dipping volley was well kept out by Kenny before another goalbound effort was blocked, this time by Kyle Bartley.
But the hosts, who had been a non-entity as an attacking force, made Swansea pay for their wastefulness by taking the lead in the 62nd minute.
Mark Yeates fed the ball to Cresswell, who controlled the ball on his chest and hooked home an impressive volley from eight yards.
Swansea had a chance to restore parity just minutes later when Pratley put Mark Gower through on goal, but the midfielder dragged his shot wide of the post.
The visitors pushed for a leveller but could not fashion any more clear-cut chances, and United put the game to bed when Cresswell's effort deflected in off De Vries, to leave Swansea's play-off dream out of their hands.