Per team policy, the Celtics didn't disclose the terms of the deal. League sources told ESPN that Brown agreed to a four-year, $115 million contract.
Brown, 22, is part of the NBA's 2016 draft class that is eligible to sign rookie extension deals before Monday's 6 p.m. EDT deadline.
Brown's agent, Jason Glushon, and Celtics general manager Danny Ainge negotiated most of the contract's details over the weekend in Boston, according to ESPN.
"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. "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, is entering his fourth season with the Celtics. He has career averages of 11.2 points, four rebounds and 1.3 assists in 24.4 minutes per game across 222 contests (115 starts).
Brown was a starter for Team USA in the 2019 FIBA World Cup this summer.