The sides exchanged first-half goals from Patrice Evra and Michu, but despite enjoying periods of dominance in an open and progressive game neither could fashion a winner.
United's disappointment will be the greater as they recovered from a tepid first-half display to dominate the second period and create enough chances to win the game.
But with Ashley Williams, Michel Vorm and Chico Flores all in outstanding form in the home defence, and Wayne Rooney turning in one of his poorest displays of the season, they could not find a way through.
For Swansea, the point was greeted like a win by the home fans and was thoroughly deserved, as they again demonstrated a faith in possession football against a top-four side, with midfielders Kemy Augustien, Wayne Routledge and Jonathan De Guzman arguably its outstanding performers.
United were boosted by the return to the starting XI of Nemanja Vidic, whose recovery from a knee injury allowed him to lead the side out in a league fixture for the first time since September 15.
With Rio Ferdinand rested, Vidic partnered Johnny Evans, but if Alex Ferguson was hoping the Serb's return would tighten up a defence that has leaked goals all season he was to be disappointed.
The rust gathered in his long lay-off was still evident as he initially struggled to deal with the threat posed by Swansea's Michu, a revelation and the league's leading scorer.
Twice in the opening quarter-hour Vidic was caught leaving the Spaniard with good shooting opportunities.
These early chances were indicative of the half that followed. Having set 11 men behind the ball in vain a week ago at Tottenham, Swansea were undaunted on their own patch even against the league leaders.
For their part, United continued as they have played the season thus far; open and jittery when under pressure, but always carrying a goal threat.
So it proved on 16 minutes when Patrice Evra flattered the visitors with the opening goal. It came from a corner, the third time the defender has scored with his head from a Robin van Persie delivery this season.
Vorm, returning to the Swansea goal after eight weeks out, had conceded the set-piece when saving smartly from Ashley Young, but was left exposed as Ben Davies failed to track Evra and his glancing header crept in at the far post.
If the goal was a shade more than United had merited it did not deflate the home side. With Leon Britton metronomic in midfield and combining with Routledge, De Guzman and Augustien, Michael Laudrup's side dominated possession and chances as the game unfolded.
Their reward came in the 29th minute, Michu finishing off a neat move to score his 14th goal of the season, 13 of which have come in the league. Not for the first time United were opened up by Swansea's midfield trio, Routledge turning Augustien's pass into the path of De Guzman, in space behind Evra.
The Dutchman's shot from the angle was saved by De Gea but fell to Michu who tapped in from five yards.
United only threatened sporadically, with Ashley Williams enjoying another fine afternoon at the heart of the home defence. Even so, they should perhaps have led at half-time, Rooney shooting straight at Vorm from 12 yards after Young picked him out from the byline.
With the pattern unchanged in the opening stages of the second half, Ferguson's response was to withdraw Antonio Valencia, throwing on Javier Hernandez to partner Van Persie, and moving Rooney to the left.
The change did little for Rooney's contribution -- a miserable afternoon saw him booked and replaced by Ryan Giggs with a dozen minutes remaining -- but it energised United, who began to find the rhythm that had hitherto escaped them.
Twice they hit the woodwork as they pressed for a winner. The first came from a Rooney cross, flicked on by Van Persie over Vorm, who could only watch as it bounced to safety off the top of the bar.
The goalkeeper's intervention was crucial to the second as he made a reaction save to push a Michael Carrick header onto the bar.
That chance had followed a flare-up between Van Persie and Williams that saw both booked, and the Dutchman perhaps lucky to escape with a yellow.
Fouled on the edge of the Swansea box, Van Persie was struck as Williams hacked the ball clear. Furious, the forward lunged for the Swansea captain.
With 10 minutes to go United's frustration grew as their claims for a penalty were denied by referee Michael Oliver.