Robin van Persie and Danny Welbeck both scored twice as Moyes recorded a 4-1 success in his first league fixture in charge of the champions on Saturday.
Two goals in as many first-half minutes set United on their way, and even when Wilfried Bony made it 3-1, the visitors continued to press.
They were rewarded with a fourth goal in injury time as Welbeck lifted the ball over Michel Vorm, and Moyes admitted that he had felt the weight of expectation prior to kick-off.
"You always want to win your (first) game - when you're manager of Manchester United your job is to win the games. I was made well aware of it," he told MUTV.
"The players have done it today and did really well.
"I thought Swansea had a lot of the ball, anybody who comes here will be made to work really hard. We had to work hard as we had periods where we didn't have a great deal of the ball.
"We limited them to very few opportunities and we made incisive, telling passes and got in for opportunities.
"More importantly, they were really good finishes by the forward players and a brilliantly taken first goal by Robin (Van Persie).
"We could have played better at times in the game overall, but are more than happy winning here 4-1."
Moyes refused to get too carried away with the opening-day victory and insisted that attention was already moving to their next outing - against Chelsea on August 26.
"In the dressing room, it's quiet as they tell me it's what they do here," he added. "Move on to the next game and that's what I'm going to do as well."
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