Callaway was given a three-year contract and will replace Terry Collins, who stepped down as manager to become a member of the team's front office. New York finished 70-92 this season.
Mets hitting coach Kevin Long, Seattle Mariners third base coach Manny Acta, Chicago White Sox bench coach Joe McEwing and Houston Astros bench coach Alex Cora also reportedly interviewed for the job. Cora was named the Boston Red Sox manager on Sunday.
Callaway, 42, joined the Indians in 2013, and he is given much of the credit for building the Cleveland pitching staff into one of baseball's best.
Cleveland led the majors in ERA (3.30), strikeouts (1,614), shutouts (19) and complete games (seven) this season. It finished third with an opponents' batting average of .236.
He also had a knack for keeping pitchers healthy, and that is a significant factor for the Mets. Injuries to pitchers Matt Harvey, Noah Syndergaard, Steven Matz and Zack Wheeler affected New York's chances to compete for a postseason berth this year.