Turkey forward Cenk Tosun had put the visitors in front after 20 minutes with his first goal in English football.
The Clarets, though, produced a second-half recovery to mark Sean Dyche's 250th game in charge with a hard-earned win. Following some fine saves from England keeper Jordan Pickford, Ashley Barnes equalised in the 56th minute, before striker Chris Wood headed in from a corner 10 minutes from time.
Sean Dyche, celebrating his 250th game in charge, was delighted as a “winning performance" that earned first win in 12 league games.
He beamed: “It's a fantastic way to bring up 250 games. We've turned a game around to win for the first time in the Premier League, with a winning performance. And I thought it was a winning performance.
“A number of things worked well for us today. The way the lads performed was key, but the lack of disappointment in going one goal down was evident.
“Everyone stayed calm, including the crowd, which is an important thing. We're on a tough run, which we've never denied and that gave the players the freedom to go and play and in the second half, in particular, I thought we were tremendous.
“To turn a game around in that fashion was very pleasing and very deserving, I must say."
Dyche said: “That pass from Matt in the second half was top, top drawer. To find that gap was one thing, but then the weight makes it one of the next passes I've seen in a long time.
“Now we have 40 points and I want us to take on the remaining part of the season with that touch more freedom, because that psychology of having that total on the board is a good marker."