"It very much was a Manchester United display," Moyes told the BBC. "We were clinical in attack and I am very pleased.
"This is a tough place to come and we won well in the end."
Robin van Persie and Danny Welbeck scored two goals each to subdue a feisty Swansea side whose high-tempo passing game made life difficult for United in the opening half hour at a rainswept Liberty Stadium.
"Today was my first game at Manchester United. I was excited but it's just work to me, it's the same as I've done at Everton over the last 10 years or so," said Moyes whose team failed to shine in their pre-season games before beating Wigan Athletic 2-0 to win the Community Shield on Sunday.
"Robin van Persie scored two brilliant goals. When he pulls the trigger it's a goal all the way."
Van Persie continued the superb form he showed last season following his transfer from Arsenal.
BOUNCING BALL
The Dutch striker controlled a bouncing ball after 34 minutes and cleverly hooked it into the net with his unfavoured right foot to put United in front and Welbeck tapped in the second from close range soon after.
Moyes sent on Wayne Rooney as a substitute on the hour and the England forward, whose future is the source of much media speculation, was involved in the move which set up the third goal 18 minutes from time.
Van Persie was freed to advance on a back-pedalling Swansea defence and, ignoring options either side, he lashed a fierce left-foot shot high into the net from the edge of the penalty area.
Swansea deservedly pulled one back through a neat finish by substitute Wilfried Bony but United finished the game on the front foot and added a fourth goal in stoppage time.
Rooney's deft pass released Welbeck down the right channel and the England striker, who scored only two goals for his club last season, dinked the ball high over goalkeeper Michel Vorm from a narrow angle and it nestled in the far corner of the net.
"I thought Danny Welbeck was terrific," Moyes told Sky Sports. "He sometimes doesn't get the praise he deserves because of some of the other strikers we have. But that was a great finish."
Swansea manager Michael Laudrup also praised United's clinical finishing.
"For the first 30 minutes we were very competitive and at moments we dominated," he said. "You can come back from that but when they scored a second, against a team like Manchester United - it's difficult.
"The difference was in the finishing but that second goal really destroyed us in many ways and we never really recovered from that.
"Three of their four goals were outstanding but you can't say we played badly especially in the first half," added Laudrup.
(Editing by Tony Jimenez)
