Thierno Barry struck on the hour mark for the 1-0 win at Villa Park.
"I've got to say, I came here a bit worried about what we had available to us, probably more to do with what we were missing and how we'd go about it, but we've worked all week," Moyes told evertontv afterwards.
"We're really sore from the last week, and I've been struggling to show my face because I want us to be doing well and I felt like we hadn't done that well after going out of the Cup.
"I said to the players, 'Let's give the supporters something today', and with the players we had available – and it's no disrespect to them – it's an incredible result against a team that's challenging for the top of the league. They might have been missing one midfield player but we were missing four or five regulars, which has shown in our recent performances – but it didn't show today.
"I think this will give us a huge boost. We've been waiting to get a lift and it's been a difficult sort of three, four weeks with people out and losing Kiernan Dewsbury-Hall, Charly Alcaraz, and even behind the scenes today we had one or two problems going into the game and we didn't know if (those players) were going to be available or not, so we've had to cope with all of that.
"It's not made the past week or so very nice because we're coming to Aston Villa, who have won 11 games on the bounce, looking to try to get a result – but there's something about Everton, something about the resilience, something about the Club, something about the supporters that can make those magic moments happen."