Substitute Rose struck five minutes from time when Clarets goalkeeper Brian Jensen could only touch Scott Golbourne's overhit cross onto the crossbar and back into the 19-year-old's path.
The visitors were forced to play the majority of the tie a man down after midfielder Brian Stock was shown a second yellow card in the 37th minute for a cynical shirt pull on Jim O'Brien.
Flitcroft said: "We deserved the result, the lads deserved the result, there's no doubt about that. The work that the boys have put in...
"I was made up for Danny Rose getting the winning goal, he's somebody who will score goals for this football club and he's got a real important one.
"This boy, once he grows into his body, he's going to be some player and some goalscorer for this football club.
"We've protected him and kept him where he's at. And Danny Rose will be all right for this experience."
He added: "I'm delighted for the boys, they've worked hard, they've fought well and they've stuck to the task. They stood up to everything Burnley threw at them."