Both sides struggled to handle testing conditions during a first half which was halted for 10 minutes due to a heavy hail storm.
It was, however, to be United who edged themselves in front just past the hour mark through Ashley Young, and they never looked.
Moyes admits that strike was the turning point in the tie, with skipper Patrice Evra going on to make the game safe.
The United boss told Sky Sports: "I think the first goal tonight, whoever got it, was going to make a big difference.
"It was a tight game, we had some good play at times but we hadn't created too many chances, but I don't think Stoke created too many either. I thought that once we got the first goal we were in control. When we got the second goal, we went on and saw the game out well.
"The conditions were a leveller, particularly in the first half it was really difficult for the players to play. The wind was blowing all over the place. Players will tell you that to play in heavy wind is as bad as anything."
United have been paired with Sunderland in the last four and Moyes hopes his decision to take the competition seriously will pay dividends as he chases major silverware in his first season at the Old Trafford helm.
He added: "Manchester United Football Club are used to winning trophies, so it's no surprise to them and it's nothing different. It's business as usual but we are all delighted that we're in the semi-final.
"Going into the last couple of rounds it looked like the top teams were going to be in and around the semi-finals, but some big results tonight and last night mean it has changed around a little bit.
"But Sunderland have just had a really good result against Chelsea and we are going to have to play well over the two legs to get to the final."
Wayne Rooney missed the Stoke trip with a groin injury but Moyes revealed he could still be fit to face West Ham on Saturday.