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.
Stevens led Butler University to two national championship games against Duke and Connecticut, never winning fewer than 22 games.
He became the youngest coach to reach the Final Four since Bob Knight in 1973.
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.