The 27-year-old German point guard had been a key part of the Lakers side which reached the playoffs last season, bowing out 4-2 to the Phoenix Suns in the first round.
Schroder averaged 15.4 points, 3.5 rebounds and 5.8 assists per game during the 2020-21 NBA season, but is the latest in a series of off-season changes at the Lakers including the arrival of Russell Westbrook.
Schroder wrote on Instagram: “I’m proud to announce that for the 2021-22 season I’ll be playing with the Boston Celtics!
“This is one of the best franchises in NBA history and it will be a (sic) honour to put on the green and white and do what I love!
“I’m going out there every night and leaving it all on the floor for the city!”
ESPN has reported that the one-year deal is worth $5.9m for the unrestricted free agent.
Schroder had declined a lucrative extension offer from the Lakers during the regular season.
The German joins Jayson Tatum and Jaylen Brown at the Celtics, who have allowed point guard Kemba Walker to join the Oklahoma City Thunder.
The Lakers have signed Westbrook along with veteran Carmelo Anthony this off-season along with Trevor Ariza, Kent Bazemore and Dwight Howard.
Andre Drummond, Kyle Kuzma, Kentavious Caldwell-Pope and Montrezl Harrell are among those to leave.