The relief was clear inside the iconic stadium after Antonio Valencia and Danny Welbeck scored second-half goals to revive a season that was in danger of derailing under new manager David Moyes.
The win not only keeps seventh-place United in touch with the top six but will go some way to restoring confidence after a chastening start to 2014 that saw the team defeated three times in six days.
Valencia swept home the first goal from close range less than two minutes after the break, before Welbeck glanced in a deft finish in the 59th for his sixth goal in his last six league games.
Not since 1961 _ the year the Berlin Wall was built _ had United lost four straight games but that was the ignominy staring at England's biggest club after being defeated by Tottenham in the league, Swansea in the FA Cup third round and then Sunderland in the League Cup semifinals on Tuesday.
Wayne Rooney and Robin van Persie were missing for the third game in a row for United because of injury while another big name was absent from the stands _ former manager Alex Ferguson.
The presence of Ferguson at both home and away games this season has been regarded by some as putting extra pressure on Moyes and his stuttering team, but the Scot was reportedly in Barbados on a family vacation. He will return from the Caribbean with United in a better state of health.
For 45 minutes, it was another ordinary performance from Ferguson's old side, which _ it should be remembered _ cantered to the league title by 11 points last season. It enhanced the feeling that finishing in the Champions League places may yet be beyond the class of 2013/14.
They weren't short of chances in the first half, though, with the excellent Adnan Januzaj whipping an 11th-minute free kick against the bar, Darren Fletcher blazing over when well placde in the 35th and Welbeck clipping wide a gilt-edged opportunity from 12 yards out moments later.
Swansea's midfielders were enjoying plenty of possession and, more tellingly, time on the ball. The visitors created few clear-cut opportunities, however, and were hit by an early setback barely a minute into the second half.
Januzaj's cross was inadvertently flicked to the back post by Chico Flores, on early as a substitute, into the path of Shinji Kagawa whose header was parried away by goalkeeper Gerhard Tremmel. Valencia was in the right place to sweep the rebound in at the near post and relief spread around Old Trafford.
Relief turned to joy in the 59th when Welbeck produced a neat flick to turn home a scuffed shot from Patrice Evra, another example of how the England striker is developing as a finisher.
Chances to extend the lead even further came and went, with Leon Britton performing heroics to clear Kagawa's shot off the line in the 76th minute.