Brown will spend the entire 2018/19 campaign at Elland Road and has become the club's sixth signing of the summer transfer window.
The 21-year-old, who has already experienced promotion from the Sky Bet Championship can't wait to get going at Elland Road.
“It feels amazing to be part of such a huge club," said Brown.
“My first ever game I watched live was at Elland Road, Chelsea versus Leeds when I was younger and ever since then I've remembered the atmosphere.
“I felt that same atmosphere when I played for Rotherham United against Leeds and I am so happy to be here.
“The fans always back the team 110 per cent and they give the team an extra boost on the pitch, they are a huge part of this club and it is an amazing club.
“The decision was easy, I've known of Leeds' interest for around 18 months now and ever since then it has been in the back of my mind.
“It is a huge club with a huge fan base, it has an amazing stadium, an amazing training ground and it is a Premier League club in a way and hopefully that is what we can become.
“I've watched every Leeds game on Sky, they have been amazing and you can see the way the manager has got them playing, that is the kind of football I want to play.
“They press high, win the ball back high up the pitch and try to score as many goals as possible, it looks fun and I can't wait to be part of it.
“I'm so happy to be part of this team and I can't wait to get out on the pitch and get playing."