Brown, 22, is part of the NBA's draft class of 2016 that was eligible to sign rookie extension deals prior to a Monday deadline.
Glushon and Celtics general manager Danny Ainge negotiated most of the deal's details over the weekend in Boston.
This is the first rookie-scale contract extension that Ainge has done since Rajon Rondo in 2009.
"Jaylen has made tremendous strides over the last three years and has become a fantastic player for us on both ends of the court," Ainge said in a statement released by the team. "He's a great person and hard worker who, at 22 years old, can score and defend against the best players in the NBA. Jaylen is a true professional who did a great job accepting his role last season, and he is a major part of our championship goals."
Brown, the No. 3 overall pick out of California, has steadily developed into one of the league's top young forwards. He's averaged 11.2 points in three NBA seasons, including 13.9 in last year's playoffs and 18 points a game in 18 playoff games in the 2017-18 postseason.
Brown was a starter for the U.S. in the FIBA World Cup over the summer.