The Toffees conceded early when Papiss Cisse netted inside three minutes for the Magpies but rallied via goals from Leighton Baines and Victor Anichebe to move to fourth on the table.
"We had to do it the hard way again, unfortunately, but that has been the way it is,” Moyes told Sky Sports.
"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."