The 33-year-old left-back, who will sign a two-year contract, is manager Neil Warnock's second summer signing following the arrival last month of Jason Pearce from Portsmouth.
The club has also announced the arrival of midfielder Paul Green on a free transfer from Derby.
The 29-year-old Republic of Ireland international has agreed a two-year deal at Elland Road and will formally link up with the club at the start of next month when his Rams contract expires.
"He's a very good professional," Warnock said of Drury club's official website. "I think he's a good defender, he's quick and I thought we needed a bit of experience in certain places, and that desire and will. And with Adam you know what you're going to get."
He added: "I'm delighted to have him. He's got the experience we need. He played 16 times for Norwich last season and he's better than a lot of full backs in the Premier League.
"He's a good lad. That's half the battle as well. It gives us a good mix because we've got young Pearcey on that side as well so we can put that to bed now and move on to a different area. When you speak to him he's full of enthusiasm and when you get released from a club you want to prove people wrong."
Drury, awarded a testimonial match against Celtic at Carrow Road in May, has made over 350 appearances for Norwich, who he joined from Peterborough for £500,00 in March 2001.
Of Green, Warnock added: "Paul is another player I've been after for a long time, but obviously he was called up by Ireland for the European Championships so we've had to be a bit patient.
"I've seen him grow from his days at Doncaster and I'm delighted to have him here.
"I've always thought he is a player who can create goals as well as score them, and he has just the type of attitude we need at Elland Road."