Budenholzer, 43, has been an assistant with the Spurs for the past 17 years and been with the organization under coach Gregg Popovich for 19 years overall. He has a long-term relationship with Hawks general manager Danny Ferry, who played for Budenholzer and worked with him as an executive with the Spurs.
"He has an incredible basketball acumen and has a keen awareness of the league and what it takes to be successful," Ferry said in a statement released by the team. "His experience and four championships over the last 17 years provide a tremendous foundation for his leadership of our team."
The Spurs allowed the Hawks to interview Budenholzer during the playoffs and then gave approval to negotiate a contract with him after they advanced to the NBA Finals with a victory over the Memphis Grizzlies on Monday night.
The Hawks announced that Budenholzer will remain on San Antonio's coaching staff until the conclusion of the NBA Finals. The team has scheduled a news conference for Budenholzer on Wednesday in Atlanta.
"I have been extremely fortunate to be a part of the San Antonio Spurs organization for the last 19 years," Budenholzer said in a statement released by the Hawks. "I knew it would have to be a tremendous situation for me to leave and clearly coming to Atlanta as the head coach of the Hawks is perfect for me."
The Hawks lost to Indiana in the first round of the playoffs. It was the team's sixth straight postseason appearance but Ferry is expected to continue his dramatic makeover of the team after trading Joe Johnson and Marvin Williams in his first year as general manager.
The Hawks are in flux with their roster, as they have just three players under contract for next season and could have in excess of $40 million in salary-cap space. Atlanta is expected to chase top free agents such as Dwight Howard and Chris Paul as well as their own free agents, Josh Smith and Jeff Teague.
The Hawks believe Budenholzer, who has interviewed for several jobs in recent years and also a candidate in Detroit this year, will be able to operate no matter which way the roster decisions go.
"Building a roster that has terrific potential because of the existing core and the ensuing flexibility presents a rare and uniquely positive opportunity," Budenholzer said in a statement.
The Hawks are expected to cut ties with head coach Larry Drew, who had the job for three seasons. Drew's contract is up June 30, and he's reportedly having a second interview with the Milwaukee Bucks for their head-coaching job this week.
The Hawks' new coach is looking for his fifth NBA championship ring this year. The Spurs won NBA titles in 1999, 2003, 2005 and 2007.
Budenholzer began his time with the Spurs as the team's video coordinator for two years before being named an assistant coach. He will leave the Spurs with Popovich's blessing.
"I couldn't be happier for Mike for many reasons," Popovich said. "As anyone who's been part of this program knows, he has been more of a co-head coach than an assistant for a long time. His knowledge of the game as well as his ability to teach and develop relationships with players are all special. I will miss him a great deal both professionally and personally and am confident that he and Danny will make a great team as the future unfolds."
"Mike Budenholzer has been extremely important to our success in San Antonio," Spurs star point guard Tony Parker said. "He is a great coach. I think the Hawks made an excellent choice and I'm very happy for Coach Bud."
ESPN's Chris Broussard and The Associated Press contributed to this report.