Gray has signed a three-year deal with the Clarets which, subject to paperwork, beats the £3million paid for Steven Fletcher and George Boyd in 2009 and 2014 respectively.
The 24-year-old turned down a move to Bristol City on Thursday after a £9million fee had been agreed between the two clubs.
Gray told Burnley’s official website: “It was quite an easy decision in the end. As soon as Burnley’s name came up I definitely wanted to speak to them and find out what they had to say.
“It was quite straight forward after that. I spoke to my agent and (former Brentford manager) Mark Warburton and his assistant too.
“They have a good relationship with the gaffer, so I got another honest opinion from someone I trust and they sang the praises of Burnley Football Club and especially the manager.”
Sean Dyche’s men are without a win in three Championship games so far this season and also crashed out of the Capital One Cup to Port Vale.