Jim Sundberg led off the ninth inning with his third hit -- a bloop single -- moved tothird on first baseman Bill Buckner's error and scored on Willie Wilson's single in Boston.
The Red Sox had tied, 4-4, on Rick Miller's pinch single with two out in the eighth.
Kansas City needed only four pitches to score the winning run against Bob Stanley (0-2). Sundberg, who had had two doubles, poked his opposite-field single to right, then took third when Buckner fielded Onix Concepcion's bunt and threw wildly to second. Wilson's grounder then caromed off second baseman Marty Barrett's glove. The official scorer gave Barrett an error, then more than an hour later changed it to a hit.
Winner Dan Quisenberry (2-2) was the third of Kansas City's four pitchers. He gave up an unearned run in the eighth as a result of his wild pickoff throw.
Kansas City had tied the game, 3-3, in the fifth on Sundberg's double and Wilson's single and taken a 4-3 lead in the seventh on George Brett's two-out single against Bobby Ojeda, who replaced starter Al Nipper after the Royals had put runners on first and second with two out.
Kansas City starter Danny Jackson allowed six hits and five walks before Joe Beckwith relieved in the seventh.