United came through a potentially tricky opening test at the Liberty Stadium with flying colours as two goals apiece for Robin van Persie and Danny Welbeck sealed a 4-1 victory.
Moyes admitted it was a relief to emerge with three points but insisted there was still room for improvement.
Asked if it was a dream start to his reign, Moyes told Sky Sports: "It is because you always want to win your games and as manager of Manchester United your job is to win games.
"Swansea had a lot of the ball and they passed it very well and are very well managed under Michael Laudrup, but we limited to them to few opportunities.
"They were quality goals. We could have played better at times but I am more than happy with winning 4-1.
"It is one game in the season and it is just the start for me. There is a long way to go but there's no big celebration.
"We've done a good job because this will be a really hard place for teams to come and play, so to come here and win was good."
Both of Van Persie's goals were excellent strikes and Welbeck also completed the scoring with a sublime chip, but it was the England striker's first from close range which most pleased Moyes.
He said: "I like centre-forwards to score tap-ins inside the six-yard box and if you are going to get a lot of goals you are going to have to get a few of them, so it is a good start for Danny."
Wayne Rooney's future has been one of the big talking points of the summer at Manchester United but Moyes has always insisted he is not for sale and brought the striker on for the final half hour against Swansea.
He received a warm ovation from the United fans and played his part in setting up United's final two goals.
Moyes said: "He got 65 minutes in the game against Scotland the other day and I felt he could do a job if he came on.
"He's not ready yet physically because he's not done enough training, but he came on and a made a great run for Robin's goal and played in Danny for his goal, so it will be good for Wayne to get those minutes.
"I think people know good players and Wayne is a good player. Everyone wants those players in the team and we are delighted to have Wayne playing."