The shot stopper was out of contract after his deal at Crystal Palace expired this summer.
The Clarets have secured the Welsh star on a two-year deal, which includes the option for another year.
Hennessey will be backup to Nick Pope, while he will give youngster Bailey Peacock-Farrell a chance to go out on loan.
"Bailey and Will are fantastic goalkeepers in their own right, but in terms of Premier League they are still relatively inexperienced," said goalkeeping coach Billy Mercer.
"Wayne has bags of experience but it's not just a given that he will come in and be in a certain position. He's going to have to work hard for it which he's proved over the years is what he does.
"He works hard at training and works hard at his game. He still got a desire to play for his country, which is really good.
"He has a passion because he wants to get to 100 caps, which is incredible when you look at the goalkeepers who have played for Wales before him. I am excited and it is strengthening the goal keeping department that little bit more."
Mercer added: "We've done a lot of homework finding out about his background and he is everything you want at a football club.
"He is very supportive if he's not playing, which is vital because we pride ourselves on the team spirit we have here.
"He's a great addition to what we already have as a great group of goalkeepers."