Agent Kevin Bradbury of BDA Sports completed the deal Wednesday with Boston GM Danny Ainge.
Johnson, one of the best forwards on the market, departs after six seasons with Toronto. He gives Boston a long, athletic and experienced dimension that it needed in its frontcourt.
Johnson averaged 9.3 points and 6.1 rebounds in 75 games for the Raptors last season.
The Celtics can't officially sign Johnson until July 9 when the NBA's free-agent moratorium ends.
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