The Red Sox greeted Royals starter Alec Marsh (0-8) with four extra-base hits in the first inning, grabbing a 3-0 lead. Triston Casas' two-run home run extended his hitting streak to a career-best nine games.
Enmanuel Valdez drove in a pair of runs with a two-out single as Boston expanded its lead to 6-1 in the third, chasing Marsh after 2 2/3 innings. He allowed six runs on seven hits and two walks.
Rafael Devers led off the fourth with his second double of the game as the Red Sox took an 8-1 lead and Trevor Story's RBI double in the sixth made it 9-1.
Tanner Houck (4-8) exited after hitting Matt Beaty with a pitch, loading the bases in the sixth as Kansas City batted around, cutting the deficit to 9-4.
Houck completed five innings, allowing four runs on seven hits and a walk, fanning five to earn his first win since April 20.
Nelson Velazquez's bad-hop RBI double got Kansas City on the board in the second. Velazquez had two hits and two RBIs.
MJ Melendez capped the scoring with a leadoff homer in the ninth. He reached safely five times, with three hits, two walks and two RBIs.
Nick Pivetta earned his first save of the season and third of his career, striking out five over the final three frames.
Alex Verdugo was a home run shy of the cycle Saturday. He led off the game with a triple, then doubled with two out in the second and added a fifth-inning single. Verdugo has six multi-hit games in his last 12, batting .357 with eight extra-base hits.
Beaty was hit three times by pitches, setting a Royals' club record.
—Field Level Media