With the score locked at 1-1, Forest skipper Morgan Gibbs-White won and converted an 89th minute penalty for the winner.
Dyche said afterwards: "We know we played a lot better than that with the ball, but you have got to find a way to win. You have to earn the right, and we have done.
"A very edgy game. Monday night, no other football on and even bigger with all the noise around it.
"We wanted to come out the right side of it and we have. Very pleased with not only the performance, but the grit and determination of the team to find a way to win."
He continued: "Once we got the first one, I thought we were asking a lot more questions of what we did in the first half.
"Second half, we asked a few more questions and made it a mixed game and made it a bit more awkward for them.
"I thought Dominguez was fantastic. His team play was out of this world. He is good-natured to go win it, and to go press. A very good team player."
On pressure building after four successive losses, Dyche added: "Don't forget we have lost four on the trot and noise built up quickly. With other teams, there didn't seem to be that type of noise.
"We have got to go again. That is part of the next mentality and the growth."