Crawley, who had reached the fifth round in each of the last two seasons, made the perfect start at the Broadfield Stadium with Nicky Adams' 25-yard strike.
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," McDermott said 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.
"We made a few changes but we put out a team we thought could win and that is what happened. It has turned out to be an okay day. I am pleased to be in the next round."
McDermott hopes the victory can help boost Reading's bid for top-flight survival, a campaign which will continue next week against West Brom.
"It was important we won and we have done that," he said. "We are in the results business and we need to get a result against West Brom."