Trailing to a Steven Naismith goal at half-time, McDermott's side turned it round thanks to an Adam Le Fondre double.
The striker made amends for a first-half miss with a match-winning contribution, his second goal coming from the penalty spot.
McDermott said: "Adam Le Fondre scores goals, and that's what he's done. He nearly scored another one in the first half when he went round the goalkeeper. I think he's had three chances and scored two.
"I'm delighted for the players. They've deserved that and we've deserved to get more points, but the only thing relevant is the result and we've got that here.
"Everton are a great side, they've players worth an awful lot of money and they're an established side - David Moyes has been there 10 years now.
"We had to work ourselves into the game, we scored a really good goal and then got the penalty. I just told them to get the tempo up - play the way we play.
"This is never an easy place for anybody to come and, although Everton played well in the first half, we didn't play to the levels we're capable of. But we really picked up, and the fans were fantastic."