Stevens, 36, has spent the past six years as head coach of Butler University and will replace Doc Rivers, who was released from his Boston contract nine days ago to take charge of Los Angeles Clippers in exchange for a future draft pick.
"Brad and I share a lot of the same values," Celtics president of basketball operations Danny Ainge said in a statement. "Though he is young, I see Brad as a great leader who leads with impeccable character and a strong work ethic.
"His teams always play hard and execute on both ends of the court. Brad is a coach who has already enjoyed lots of success, and I look forward to working with him towards Banner 18."
Stevens led Butler University to two national championship games against Duke and Connecticut, never winning fewer than 22 games.
He led the Bulldogs to a 166-49 record, including a 12-5 record in the NCAA men's basketball championship, and became the youngest coach to reach the Final Four since Bob Knight in 1973.
"Our family is thrilled for the opportunity given to us by the leadership of the Boston Celtics, but it is emotional to leave a place that we have called home for the past 13 years," Stevens said in a statement release by Butler University.
"We truly love Butler University and Indianapolis, and are very thankful to have had the opportunity to celebrate so many wonderful things together. What makes Butler truly unique is the people that we have been so blessed to work with."
Butler president James M. Danko said: "In keeping with The Butler Way, Brad has given his talent to our university with exceptional generosity, integrity, and humility.
"We have done everything we can to keep Brad here at Butler; however, the Celtics team has offered Brad and his family a unique opportunity with which no university can compete."
Stevens will take over a Celtics team that finished third in the Atlantic Division last season with a 41-40 mark after winning the crown five years in a row and lost to the New York Knicks in the first round of the playoffs.
During Rivers' nine seasons at the helm, Boston went 416-305 in the regular season and 59-47 in the postseason, winning a NBA championship in 2008.
Though they enjoyed a long run of success after they assembled a 'Big Three' when they acquired center Kevin Garnett and shooting guard Ray Allen to join forward Paul Pierce, the nucleus of the team has been dismantled over the past two years.
Allen left Boston to join the NBA champion Miami Heat last season and both Garnett and Pierce are expected to join the Brooklyn Nets in a block-buster multiple trade later this month with the Celtics set to gain several players along with three first-round draft picks.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Greg Stutchbury)