As it is, that depressing start to 2014 has been halted, even if Moyes accepts there is still plenty of work to do.
"From the start of the second half I thought we played very well," said Moyes.
"I hope it is going to improve an awful lot on that but you have to start somewhere and that was more like it.
"We scored two goals, should have had a few more and I can't remember Swansea having a chance hardly."
Although United started brightly, their failure to score led to them becoming anxious.
It took Moyes' half-time decision to switch Adnan Januzaj, who had started in a central position, with Kagawa, who began on the left, to change the course of the game.
Januzaj was outstanding and the 18-year-old provided the cross from which Valencia eventually scored after Kagawa's header had been saved.
And he was also involved in the move that ended with Welbeck flicking in his sixth goal in nine games.
"Adnan has been like that since day one," said Moyes.
"I speak to him a lot and he tells me he can play anywhere.
"He is not a nervous boy. He needs teaching and understanding because he is still learning the game but his natural talent and ability is up there with the best. In time he will prove that."
Moyes rejected stories that striker Robin van Persie could be sidelined for another six weeks as he struggles to overcome a thigh problem that has already kept him out for a month.
"I don't know where that has come from," said the Scot. "That would be wrong information as far as I am concerned."
And at the end of a tortuous few days, Moyes insisted he was handling the pressure with some degree of comfort.
"I sleep great every night," he said. "I am not nervous at all."
- Sports & Recreation
- Shinji Kagawa
- Danny Welbeck
- Antonio Valencia
- Chris Smalling
- Adnan Januzaj