Hal Robson-Kanu put the Royals in front in the 14th minute before Leroy Lita pulled Boro level with a 29th-minute strike.
Ian Harte and Shane Long both netted braces to secure McDermott's side a well-earned victory, although Lita made the scoreline look a bit more respectable for his side with a 72nd-minute penalty.
McDermott said: "You can say the right things about the need to avoid a hangover after a game like that but you have to go out and do it and we proved we were right for this game by starting well.
"At half-time I felt we need to up the tempo a bit but in the end we were comfortable winners. It was really satisfying on the back of the result against Everton. We made sure we weren't flat and even had a few one-on-ones that could have brought more goals.
"We had really good support from the fans. I think it was our fourth-highest attendance of the season. We keep talking about the FA Cup and it's been incredible for us in the past two seasons. It's like buses. You wait 83 years for one and then two come along together."