The emotion released at the end was as if Manchester United had sealed a 21st title, not beaten a Swansea City side without a league win since 4 December.
These are changing times for the champions under David Moyes, who had overseen three consecutive losses yet whose relief at this result allowed him to reject any doubts about taking United to Chelsea on Sunday.
"What? With the players I have got here at this football club?" he said. "I have no fears whatsoever with the players I have got. I wouldn't really care who plays in my team because I believe in them all. I believe in what this club stands for and they will go and take whatever challenge … and throw it forward.
"We will try and push on and try and continue to make progress. I don't think we played that badly in the last three games but we didn't win.
"Today we played OK and got the points and hopefully we can continue that." Two of the defeats were in the League Cup and FA Cup, meaning that this win was a fifth in six league outings for United.
"There you go," said Moyes. "It is amazing how you can turn things around. Five in six. If you had said that maybe six weeks ago you'd win five out of six, I think most people in the public and the media would say that is not bad. Six out of six [would be better] but five is not bad."
Yet Rafael da Silva is clear that it is down to the squad and not the manager to do the convincing.
Asked about the atmosphere in the dressing room, the right-back referred to the departed Sir Alex Ferguson, who was not at this game, and said: "We are together. We knew there would be a change. The manager had just left after 27 years. It was always going to be hard. We have to take the mentality David Moyes has given us and use it on the pitch."
Da Silva accepts there has been upheaval. "It is quite a big difference. David Moyes has a strong mentality. He is doing his own job. We have to use that in a positive way," he said, before saying that the focus should be on the players.
"I agree 100%. We are the ones who have to take the responsibility. We have to show, like we did in the second half. I don't know why everyone is looking at the manager. It is the players who have to do the job on the pitch."
Darren Fletcher expressed dismay at the view that Ferguson's presence at matches is affecting results. "It gets us angry because we know it is not true," the midfielder said.
There had been four home losses in four weeks to make a total of five this season at United's home ground and nerves were evident during an insipid first half.
At the break Moyes swapped Adnan Januzaj from his No10 berth with Shinji Kagawa on the left and the youngster and Japanese each prospered. It was Januzaj's cross that came to Kagawa and, after his shot was repelled by Gerhard Tremmel, Antonio Valencia scored.
Danny Welbeck's ninth goal of the season came when Januzaj intercepted a Tremmel throw and, from the ensuing play, the striker flicked home.
The injuries to Wayne Rooney and Robin van Persie have handed an opportunity to Januzaj, at the age of 18, to lead the side. He took corners and free-kicks and Moyes is impressed enough to bracket him with a teenage Rooney, whom he managed at Everton.
"He is in Wayne's category. He is in that territory definitely because of the ability he has as a footballer," said the Scot.
"He is slightly different. He is a real calm boy. He looks after himself in a great way. His father has really helped him a lot. He practises every day and he has that little bit of arrogance that all the top players have."
Rooney is in Egypt with his family, recuperating from an abductor injury. Whether he can return to face Chelsea remains unclear.
"A chance," was all Moyes said, though he rates the forward's prospects better than those of Van Persie, who has a thigh problem. "Less of a chance," said the Dutchman's manager.
"We are hoping it can change. I cannot tell you exactly how long it is going to be. I don't want to say yes if it is not and I don't want to say no if it could be. I don't know who wrote [he would be out] six weeks, but it is nonsense."
Man of the match: Adnan Januzaj (Manchester United)