First-half goals from Paul Scholes and Ashley Young were all the hosts could muster as they tried to respond to their rivals' victory over Newcastle.
Barring a freak United result against Sunderland, City will win the league with victory at home to QPR next weekend.
Had they enjoyed a bit of good fortune in front of goal, the Red Devils might have made inroads into City's superior goal difference. Wayne Rooney had a volley turned away by Michel Vorm, Javier Hernandez wasted two close-range opportunities and Chris Smalling sent an instinctive header wide as the hosts set up camp around the Swansea penalty area.
Scholes claimed the opener after 28 minutes, flicking home a Michael Carrick shot that was bound for the opposite corner. It was the 37-year-old's fourth Premier League goal since he came out of retirement in January, his highest haul for five years.
Young got the second four minutes before the break, latching onto the loose ball after Rooney's shot had been blocked. That Young chased into the goal to grab the ball after his goal suggested there was some purpose about United and they were quickly into their stride after the break.
Rooney's attempt to roll a pass across goal to give Young a tap-in was then intercepted by Dyer, while Hernandez was inches away from heading home before Antonio Valencia smashed a shot goalwards, forcing Vorm into another save.
Yet the loose defending that has been evident far too often this season was also apparent. Joe Allen was given far too much room to shoot on the edge of the area and when his shot was deflected by Danny Graham, it took an impressive manoeuvre from David de Gea to keep it out.
Dyer then curled a shot over when he should have at least kept his effort on target, and Swansea came close again when De Gea turned away Gylfi Sigurdsson's free-kick.
Rooney fluffed a great chance for a third and a generally disappointing day ended on a low note for Rio Ferdinand, who came off near the end after suffering what appeared to be a hip injury that could have implications for his hopes of featuring at Euro 2012.
Source: PA
Source: PA