The Toffees were forced to come from behind after Adam Johnson handed Sunderland the lead and did so through goals from Marouane Fellaini and Nikica Jelavic
The Merseyside outfit found themselves a goal down just before the break when an Adam Johnson half-volley put the visitors in front.
However, two goals in three minutes from Marouane Fellaini and Nikica Jelavic turned the game back in the hosts’ favour following the restart.
And Moyes, who celebrated his 400th Premier League game as manager with a win, expressed delight with the Toffees' spirit.
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"They made it really hard for us to get through and I actually thought after 70 minutes this might not come, but in the end it did and it was a great three points for us.
"But we had a lot of possession and just didn't create the chances our ball possession might have deserved.
"Sunderland had a couple of half chances in the first half they will be disappointed they didn't take.”
The Scot did admit he was disappointed to see his side fail to keep a clean sheet against the Black Cats and called for defensive improvement.
"The best teams can attack but the best also have top defenders and make sure the opposition don't get many opportunities," he continued.
"Over the years we've been very good defensively and maybe not so good on the other side. So getting that balance right is how you become a top team.
"I was a defender myself and I don't want us to be giving the opposition too many chances and in the main we've not done that too often."