Reading's goal came in the 26th minute - after Everton had controlled the match for long periods.
"They played the ball around well during the first 20 minutes but we kept our shape," added McDermott.
"When we did get the ball, we made sure the first pass was a good pass. Once it got to 1-0, I thought we had a real opportunity."
The Royals boss also paid tribute to the Everton fans, who applauded the visitors off the field but jeered their own team.
"The crowd was phenomenal," said McDermott.
"You very rarely see that in football and it just shows the class of a club of the magnitude of Everton. You can see the class of a club when they lose, it was the same at Liverpool."
Toffees manager David Moyes admitted his side had "let the supporters down".
"We didn't play well enough and we got what we deserved," he said.
"After coming through the Chelsea tie and getting a home draw was all you could ever ask for.
"We got that but we never came up to scratch, apart from the first 10-15 minutes when we passed the ball well.
"We didn't have enough quality, enough craft or guile to break them down at the top end of the pitch."
He also absolved Jermaine Beckford of any blame after his late arrival at the ground meant he had to start the match on the bench.
"Jermaine had been stuck in an accident, so he wasn't available to be selected," said Moyes.
"It can happen. It was a road accident and it wasn't as if the boy left late or anything. It was just one of those things.
"Anybody can get stuck in traffic nowadays."
BBC 5 live analyst David Pleat said he was delighted a non-Premier League club had reached the next round of the competition.
"I saw Reading in the last round at Stevenage and they played a good game," he said.
"I also saw them play Watford in the league and they played good stuff. It's great that the Championship has a contender in the cup."
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