United went behind early on through Peter Crouch and went in at half-time 2-1 down to Stoke, but pulled off a scrappy win with two goals in two minutes near the end of the match.
Moyes said: "I thought with 15 to 20 minutes to go the supporters were terrific - they drove the team on and they played a big part in getting the result.
"I thought in the end we had to go for it. I thought we did a lot of good things but an awful lot of really poor things that put us in the situation we were in.
"We got ourselves back in the game just before half-time only to concede from a poor, poor free-kick.
"That got us going in at half-time on a downer but I said at half-time your job is to win the game and you're more than capable of doing it. We needed the substitutes today and they made a big difference."