Referee Mark Clattenburg took the players off the field for just under 10 minutes amid swirling winds and a hail storm during the first half of the quarter-final against Stoke City at Britannia Stadium on Wednesday.
Play resumed with Ashley Young scoring his first United goal for 17 months in the second half and Patrice Evra adding a second goal as the Premier League champions secured a 2-0 win.
United will now play Sunderland in the semi-finals next month as Moyes eyes a trophy in his first season at Old Trafford.
The former Everton boss told Sky Sports: "I think especially in the first half it was very difficult for the players to play.
"It was one of the worst conditions, players will say to play in heavy winds is as bad as anything.
"The first goal was always going to make a big difference, we were in control after that and then we got the second goal and saw the game out well."
Sunderland are rooted to the bottom of the Premier League, but Moyes will not be taking Gus Poyet's side lightly after they beat Chelsea in the quarter-finals on Tuesday night.
He said: "I'm glad we are through, it will be a tough semi-final against Sunderland now.
"It's a cup game and it shows you how Sunderland are playing that they beat Chelsea. They have a new manager now so it will be tough, but we are just glad to get through."
Stoke manager Mark Hughes, meanwhile, had no complaints after the game and admitted United deserved to progress to the semi-finals.
He told reporters: "I thought we made a fair fist of it in the first half against all the elements and thought we were in decent shape.
"Our organisation was good, United weren't creating too many chances. But United, on the night, deserved to win, they just had a little bit more power and pace.
"We weren't able to ask enough questions of them in the second half."