Papiss Cisse had given the hosts a second-minute lead but Leighton Baines levelled with a stunning free-kick and Victor Anichebe scored within 90 seconds of coming on in the second half to secure victory.
Moyes said: "We had to do it the hard way again, unfortunately, but that has been the way it is.
"They've got a bundle of character, they've got great attitude. The result was right in the end.
"It was really poor defending (for Cisse's goal) and we should have dealt with it better. We expected them to come out strong and they did.
"But I have got to say, apart from that we were fantastic with our football. How we didn't go in at half time a couple of goals up I don't know."
Everton moved up to fifth with victory on Tyneside but Moyes insists he will keep his players grounded.
"You can only ever go onto the next game," he added. "We'll try and keep going, take it one game at a time; it's very simple.
"If I had a crystal ball and said we're going to have a great second half of the season, that'd be great, but I don't, so we have to prepare for every game as if it's our last and hopefully keep picking up three points."