The Clarets' run now stands at just one defeat in 13 games, keeping alive the chances of European competition next season.
Dyche said: “Everyone knows we are stretched, but the mentality has been absolutely fantastic.
“In the first half we've had to grind because they shift the ball quickly, ask questions and Popey has had to make a few top saves.
“But they also smother you so quickly, and don't allow you to get a foothold.
“We just about stayed in it, but we did say at half time that it's hard for teams, even as good as this one, to keep that going for 95 minutes.
“There was a slight drop off that allowed us to build into the game and physically, I thought we were excellent again.
“We then found a big moment again and possibly, without being too greedy, we could have found another at the end."