Crawley, who had reached the fifth round in each of the last two seasons, made the perfect start at the Broadfield Stadium when Nicky Adams' 25-yard strike put them ahead after just 14 seconds.
But two goals from Adam Le Fondre, the second from the penalty spot after Noel Hunt had put Reading ahead, secured the Royals a place in the hat.
McDermott was encouraged by the way Reading, who came into the game with just one victory from their last 10 matches, kept their composure.
"It was about coming and trying to get a result and do the basics as well as we could, which we didn't in the first 14 seconds," said McDermott with a wry smile.
"They had another chance to go 2-0 up, which would have been a real problem. It is easy to panic but we got ourselves back into the game. The reaction from our players was good.
"When we got ourselves 2-1 up it was more comfortable for us. I thought the combination of Adam and Noel was really good and Garath McCleary made a really good goal for Noel.
"It has turned out to be an okay day. I am pleased to be in the next round."
Crawley manager Richie Barker was sad to see his side's hopes of another cup adventure dashed, adding: "We are disappointed, no matter how it started, because we played well. We deserved more out of the first half.
"We have given a Premier League team a real good go today. We created enough chances to be ahead but their goalkeeper pulled off two or three really good saves."