The Toffees fell behind for the sixth successive Premier League game when Adam Johnson half-volleyed Sunderland in front in first-half stoppage time.
However, just like in their previous five matches, Everton battled back to avoid defeat, with two goals in three minutes from Marouane Fellaini and Nikica Jelavic turning the game on its head.
It ensured the home side stay fourth on goal difference and marked Moyes' 400th Premier League game in style.
"It was great, we showed great character throughout," Moyes told Sky Sports. "We never gave in, we kept knocking at the door, kept trying to score, trying everything.
"Sunderland have had a couple of half chances in the first half they will be disappointed they didn't take.
"But we had a lot of possession and just didn't create the chances our ball possession might have deserved.
"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."
Moyes also revealed his concern at his side's inability to keep a clean sheet, with Johnson's goal meaning it is just two shut-outs in the league so far this campaign.
The Scot insists he does not want to curb his side's attacking instincts, but knows they will need to tighten up at the back if they are to maintain their push for a top four place.
"The best teams can attack but the best also have top defenders and make sure the opposition don't get many opportunities," he added.
"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."