The Rams went ahead in the 19th minute thanks to Luke Moore's deflected goal but Burnley hit back when Tyrone Mears found the corner with a fine 25-yard free-kick.
A winner then arrived in the 90th minute when Jack Cork glanced home a cross from Wade Elliott from the right.
"Patience is certainly the apt word today because we had to be," said the Clarets chief.
"They wanted to allow us to have the ball at the back in the first half, very deep, and we really couldn't get going.
"That's when we changed, we did it five minutes before half-time, so the players could assess it.
"The amount of possession we had was ridiculous, yet they scored against the run of play, with a hell of a deflection, and the word patience had to be used.
"It was important we didn't get drawn into playing long balls."
The impressive turnaround helped to lift Laws's side into the play-off places as they secured only a second win in eight matches and also got within one point of tonight's opponents.
He added: "We were certainly worthy winners and I don't think anybody can take that away from us. I don't think there's any question that we have quality in the side and if we are patient we can win games.
"We showed what a good team we can be, even against a team that defended so deeply, were resolute and put a lot of men behind the ball."
Chelsea loanee Cork, who is on a season-long stint with the Lancashire outfit, will take the plaudits and Laws is keen to extend his stay with the Clarets or make it a permanent one in the not-too-distant future.
He added: "It was a great reward for a great performance from Corky. He's done a lot for us and he's enjoying his football here.
"That's something we'll talk about in January but he's here for the rest of the season. Jack knows exactly what we want to do and if there's anything we can do to keep him here we will."