The Clarets had lost five and drawn two of their first seven games to leave them second from bottom.
But Chris Wood's eighth-minute strike was enough to restore some confidence in Sean Dyche's side.
And Dyche was pleased with the victory when speaking after the game.
"We were showing signs. That's three clean sheets in the last five games and tonight we altered how we were operating in the attacking third.
"We created two or three golden chances, we take one and it gets a bit nervy because we hadn't found that first win so to get that one done will do us the world of good.
"We mixed it and asked questions and found a moment that counted. If that golden moment for Johann goes in the feel of the game changes – we have control of it. But our lads saw out the game well.
"We know they have people who can shoot from distance. Nick Pope made a good save in the first half but that one at the end was brilliant. But he is a top keeper. He'll be absolutely fine.
"We have got more people coming back fit. Unfortunately we lost Robbie Brady there. But that gives you more strength in depth. We have got a group that can challenge. We have had a tough start but that has been coming."