Budenholzer's contract is for four years, league sources told ESPN.
"We are thrilled to welcome Mike Budenholzer as the head coach of the Milwaukee Bucks," Bucks general manager Jon Horst said in a statement. "Mike has played a key role in building successful teams throughout his career. He's widely respected and has shown a special ability to teach and develop players. His leadership, basketball intellect, championship-level experience and communication skills make him the right fit to take our team to the next level."
"After a thorough coaching search, it was clear that Mike was the ideal choice as we enter into a new era of Bucks Basketball," Bucks owners Wes Edens, Marc Lasry and Jamie Dinan said in the statement. "Mike has demonstrated the ability to lead and communicate, and understands what it takes to build a winning culture."
The Bucks negotiated a deal with Budenholzer's representative on Wednesday, hours after Budenholzer attended a breakfast meeting with Milwaukee's All-Star forward Giannis Antetokounmpo and teammate Khris Middleton, league sources said.
"I'm extremely grateful to the Bucks ownership group and Jon Horst," Budenholzer said. "We have a tremendous opportunity to take the Bucks to the next level. I look forward to working with our group of young and exciting players and helping us evolve in many ways to succeed on the court. ... The tremendously supportive fans in Milwaukee and throughout Wisconsin are waiting and ready. Now it's up to us to put all the pieces together, and I can't wait to get started."
Budenholzer was Horst's top target. The other finalist for the job, San Antonio Spurs assistant Ettore Messina, was informed that he was out of the running on Wednesday, league sources said.
Budenholzer, who previously had coached the Atlanta Hawks, met with Bucks ownership on Tuesday in New York.
The Bucks wanted the two players -- especially Antetokounmpo, the franchise star -- to have a sense of the prospective next Bucks coach and encouraged the players to sit down with Milwaukee's preferred candidate before the organization extended a formal offer, league sources said.
It is a relatively unusual move in a head-coaching search, but it clearly reflects the Bucks' determination to make Antetokounmpo and fellow forward Middleton feel a part of the process.
Budenholzer flew to Milwaukee for the Wednesday morning meeting with the Bucks players.
The Toronto Raptors met with Budenholzer on Monday, but they never extended an offer, league sources said.
Budenholzer and the Hawks parted ways in April after the two sides could no longer see a path together with the franchise's rebuilding plans and new management structure.
Before the split, the Hawks gave Budenholzer permission to speak with the Phoenix Suns and New York Knicks for their head-coaching openings. He pulled out of contention for the Suns' job after several conversations with ownership and management.
Considered one of the NBA's best tacticians, Budenholzer had been the Hawks' president of basketball operations -- until surrendering front-office control with the arrival of general manager Travis Schlenk in the spring of 2017.
Budenholzer won the NBA Coach of the Year Award after a 60-victory regular season and a trip to the Eastern Conference finals in 2016. The Hawks reached the playoffs in four of his first five seasons. Atlanta dropped to 24-58 this season with the organization's mandate to shed salary and gather draft picks and future assets.
Budenholzer spent 17 years as an assistant under Spurs coach Gregg Popovich before his hiring in Atlanta in 2013. He won four NBA titles as a member of San Antonio's staff.