"Stoke have not scored many but they have not conceded many either, so we knew we would have to work hard after going behind," he told reporters.
"I was worried it wouldn't come - especially when we were 2-1 down. But I think we deserved it in the end. We kept at it and deserved to score."
Moyes also praised the impact of the United fans, although he did say that his team were poor defensively.
He added: "We had more than one or two defensive issues but our goalkeeper was fantastic. We didn't defend as well as what we would like. It may be a turning point - we know we have got to do it in style here at Old Trafford.
"The supporters really played their part today. They kept the team going when we needed them at the end."
United's next league match is away to Fulham.