Bloom, 36, will replace Dave Dombrowski, who was fired as the team's president of baseball operations in September. He was let go less than a year after winning the 2018 World Series.
The Red Sox failed to reach the postseason this year, finishing 19 games behind the AL East champion New York Yankees with an 84-78 record.
Bloom has spent 15 years with the low-budget Rays in roles including minor league operations, contract negotiations, player development, salary arbitration and international scouting. Tampa Bay had the lowest payroll on opening day, but remained competitive and reached the playoffs this season.
Bloom will become Boston's fourth hire to lead the baseball operations department since John Henry and his ownership group bought the Red Sox in 2002. Bloom is a Philadelphia native and graduated from Yale University in 2004.