The Red Sox signed Chapman to a one-year, $10.75 million contract Tuesday, Jeff Passan of ESPN.com reports.
Chapman struck out two batters in a perfect inning to earn a save against the Yankees on Friday.
Chapman struck out two in a perfect ninth inning Saturday to record his fifth save of the season in a 3-0 win over the Guardians.
Chapman (5-4) allowed one hit and struck out two in a scoreless ninth inning to earn the win Sunday against the Reds.
Chapman (3-4) picked up the win Sunday against the Diamondbacks after pitching a scoreless ninth.
Chapman picked up the save Thursday against Milwaukee, striking out one batter during a perfect ninth inning.
Chapman (1-4) blew a save and was tagged with Sunday's loss to the Mets. He allowed two runs on two hits, three walks and two strikeouts in 0.2 innings.
Chapman gave up a run on two hits in the ninth inning but still managed to pick up his third save Sunday against Atlanta.
Chapman struck out two without allowing a baserunner across a scoreless ninth inning against the Reds on Wednesday.
Chapman gave up a run on a hit and a walk in the ninth inning Saturday to record his second save of the season in a 4-3 win over the Rays. He struck out one.