Robin van Persie's double, with two also for Danny Welbeck, ensured Moyes avoided a potential banana skin at the Liberty Stadium, before Wilfried Bony netted a consolation.
When asked if the scoreline was a relief, Moyes said: "Yeah it is because you always want to win your games and as manager of Manchester United the job is to win the games. The players did that today.
"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.
"We could have played better at times but we're happy to score 4-1."
One of the big talking points ahead of kick-off is the future of United striker Wayne Rooney. Moyes gave a big hint that the England forward could be set to remain at Old Trafford this season as he started him from the bench, with Rooney claiming the assist for the Dutchman's second.
Rooney was introduced for the final half-hour and given his desire to join Chelsea, it must have been heartening for the striker to receive a warm ovation from the United fans.
It remains to be seen whether it is enough to change Rooney's view, although given United's steadfast 'not for sale' stance, that may not count for much anyway.
"He got 65 minutes in the game against Scotland and I felt he could do a job today if he came in," Moyes told Sky Sports 1.
"He's not ready yet physically as he's not done a lot of training but he came on and had a great run.
"It will be great for Wayne to get those minutes on the pitch.
"People know good players and Wayne is a good player and people want them in your team."
Van Persie's contribution was sublime, hooking in the opener before smashing home a 20-yard effort 17 minutes from time, and the Dutchman was also delighted to get off to a winning start.
"It is the first game of the league away from home. They are a great team the way they play. I love the way they play and they are very hard to play at times. It is a big win," he told Sky Sports 1.
"The first goal changed the game and we got the second shortly after so it helped us.
"The first one I had a touch and felt at the time it was quite heavy so I had to take it (the shot) as high as possible
"The first game of the season you just don't want to lose. We worked very hard so I'm very pleased."