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Hours earlier, Boston put center fielder Jackie Bradley Jr. on the 10-day disabled list because of a sprained left thumb.
Davis is expected to join the Red Sox in Cleveland on Thursday for a game between division leaders.
Red Sox president of baseball operations Dave Dombrowski said he began discussions with Oakland general manager Billy Beane last week about Davis.
He led the American League with 43 steals last year for Cleveland, and also hit a big home run in Game 7 of the World Series off Cubs closer Aroldis Chapman.
READ MORE (Boston.com)
- A's trade Rajai Davis to Red SoxSFGate
- MLB trades: Red Sox land Rajai Davis from A's following Jackie Bradley Jr.'s ...CBSSports.com
- Red Sox acquire outfielder Rajai Davis for stretch runThe Providence Journal
- Red Sox add speedy outfielder Davis in trade with A'sAOL UK
- Red Sox acquire Rajai Davis from A'sComcast SportsNet New England
- Red Sox Acquire Rajai Davis From AthleticsSports Illustrated
- Red Sox acquire speedy OF Rajai Davis from Oakland A'sBoston Herald
- Red Sox trade for outfielder Rajai Davis from the Oakland AthleticsBoSox Injection
- AL East-leading Red Sox obtain OF Rajai Davis from OaklandABC News
