Farke has replaced Sam Allardyce who left the club by mutual agreement following their relegation from the Premier League at the end of last season after he was parachuted in with only four games remaining.
"I feel humble at this time, I know the responsibility to fulfil all the expectations and I want to repay the trust shown," Farke told the club website.
"The most important thing is to create a togetherness and unity within this club again and from today onwards, I will work on it with my staff and players, and I trust our supporters will be there when we need them."
Farke returns to the English second tier after he was a manager at Norwich from 2017-2021.
The German led Norwich to the Premier League in his second season before being relegated back to the second tier in 2019/20. They won the Championship and secured an immediate return to the top flight in 2020/21.
"His record in the Championship is clear, and I am impressed by his leadership and plan to guide us back to the Premier League," Leeds Vice Chairman Paraag Marathe said.
In January 2022 Farke joined Russian Premier League side Krasnodar. However, he left the club three months later following Russia's invasion of Ukraine, without managing a game due to the winter break.
He took over at Borussia Monchengladbach in 2022/23, whom he led to a 10th-place finish in the German Bundesliga.
Farke will be assisted at Leeds by Eddie Riemer, with Christopher John and Chris Domogalla also joining the coaching team at Elland Road, having previously worked together at Norwich, Leeds said.
