The vastly-experienced 53-year-old, who has also coached his native South Africa, Australia and Pakistan, will take up the post after Sri Lanka's upcoming home Test series against the West Indies.
He will succeed Dave Houghton at the County Ground after the Zimbabwean stepped down at the end of the 2021 season.
Derby chairman Ian Morgan said: "Mickey is one of the most highly regarded coaches in world cricket and this is a key appointment for the club ahead of an important winter period.
"Throughout this recruitment process, Mickey has understood our heritage as a county, but also matched our ambition with a desire to make a positive impact at Derbyshire.
"We want to move forward from last season. We've got a talented group of players and, with a new head of cricket bringing a fresh approach, it is an exciting time for our members and supporters."
Arthur, who was interviewed for the England job in 2019, was coming to the end of a two-year contract with Sri Lanka.
Derbyshire have turned to him after a disappointing past season in which they won just six matches in all competitions. They will play in Division Two of the County Championship next year.
"This is the start of an exciting project at Derbyshire, with many young players, and I'm really excited to be a part of that and to bring my vision to the club," said Arthur.
"It's a new challenge and one which I'm eager to get stuck into."
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