Armand Traore's own goal put the Dons ahead and Ryan Lowe extended the visitors' lead before the break, with Ryan Harley and Darren Potter scoring shortly after the restart.
Jay Bothroyd and Fabio scored late consolation goals for the Hoops.
A club-record away following of 3,155 headed to west London to witness a superb performance, and Robinson said: "We felt we could come here and get a result. I understand the gulf in teams and I respect that as well.
"QPR is a wonderful football club with one of the best English managers in a long, long time so you always know the difficulties that is going to possess.
"It is nice to make it through. The players were fantastic and I thought they deserved the result.
"It was difficult for us in the first half. We had to sit off them and they have some top, top players.
"I thought the way they responded to going in front at a Premier League ground was great. They played with no fear.
"Getting into the fourth round was a little bit of history for us and now the fifth round makes it a little more special."
- Sports & Recreation
- Jay Bothroyd
- Darren Potter
- Ryan Harley
- Ryan Lowe
- Armand Traore