After a lacklustre first half, Moyes introduced Danny Welbeck at the interval and the striker went on to score the only goal of the game just before the hour to secure United's fourth straight Premier League victory.
The Scot was delighted with the impact Welbeck had on the game, but refused to get carried away as the champions moved up to sixth in the top flight.
"It's six points from two games in three days," Moyes said. "We kept going, we kept at it and we got our rewards in the end.
"I think today they (the players) found a way of winning the game. I am delighted with how they are doing but I don't think it's anything to shout about.
"Danny Welbeck changed the game. Obviously his goal made the difference but Danny's performance changed the game.
"He's a big part of Manchester United. I think Danny is a really good good player, what he now needs to become is become a good finisher, and today he done great rounding the goalkeeper."
United's sixth away Premier League win of the season moves them level on points with Everton, although Roberto Martinez's team have a game in hand.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.