United were trailing 2-1 at the break after Peter Crouch and Marko Arnautovic scored either side of Robin van Persie's equaliser.
Stoke had not won at Old Trafford since 1976 and it looked for long periods like Mark Hughes' side might leave Manchester with all three points.
But, speaking to Sky Sports after the game, Moyes was quick to praise the attitude of his players.
Moyes said: "We came good in the end. It didn't feel good in the first half. We went a goal down and got ourselves back into it, only to be behind at the break.
"It was a big challenge for the players. They did it though, showed great energy after the game earlier in the week and they kept going until the end.
"Stoke haven't scored many goals this season but they haven't conceded many either. You know you have to work hard to win the game against them and that was the case."
Moyes opted to introduce Adnan Januzaj from the bench with United trailing 2-1, and the manager says that he has got a fantastic future ahead of him.
Moyes added: "He's a really good talent and in the future he'll play an awful lot of games for Manchester United. He's an excellent player.
"Defensively though we weren't at our best today and I have to thank my goalkeeper for doing his job really well too.
"Let's hope we can keep on winning. I understand that you have to win in style here and I thought we did that in the last 15 minutes today."