Danny Valencia had two hits and drove in a pair of runs and Yonder Alonso added two hits for the A's. Josh Reddick, Khris Davis and Billy Butler also drove in runs.
''Lowrie seems like he gets a good at-bat every time out,'' A's manager Bob Melvin said. ''He's been consistent the whole spring.''
Tony Cruz homered in the seventh for the Royals.
Chris Young pitched 4 1-3 innings, giving up five earned runs and nine hits.
Joakim Soria, who gave up his first earned runs of the spring in his last appearance, pitched a perfect seventh for the Royals.
Graveman took a shutout into the seventh. He allowed four hits, struck out six and walked one.
Royals: Young was the victim of the Arizona sun as the Royals committed four errors.
''We got him past 90 pitches on his pitch count and that's the important thing,'' Kansas City manager Ned Yost said. ''A lot of those balls that found holes wouldn't be hits during the regular season. All in all, it was a pretty good outing.''
Yost also said that Young will make a ''backdown start'' in his next outing.
Athletics: Graveman felt this was an important game for him and the team.
''I've been getting incrementally better in every game and I wanted to keep that going,'' Graveman said. ''I think this was in the right direction. I was able to throw all my pitches for strikes. To win this game is big because pretty soon, they're going to count.''
Graveman was particularity happy with the development of a changeup, a pitch he worked on during the offseason and a pitch he needs.
''I think it weakens the contact and gets them off my fastball,'' Graveman said. ''I have to be able to throw that pitch to both lefties and righties.''
Royals: OF Jarrod Dyson (right oblique strain) has begun a throwing program. He's expected to return to action by mid-April.
Athletics: OF Sam Fuld returned to camp following the birth of his first child. He's been out with a sore shoulder. ... RHP Henderson Alvarez will throw a bullpen on Tuesday and is expected to throw to hitters when the team returns to Oakland either Saturday or the following Wednesday. ... Former A's shortstop and AL Rookie of the Year Bobby Crosby threw out the first pitch.
Royals: RHP Edinson Volquez is scheduled to make his first start against a major league team in two weeks and his fourth overall on Monday in Surprise against the San Diego Padres as he prepares for his opening day start. He'll be opposed by LHP Drew Pomeranz.
Athletics: RHP Jesse Hahn starts Monday's game in Mesa against the Cleveland Indians and RHP Danny Salazar.
Athletics: Optioned OF Jake Smolinski to Triple-A Nashville; Reassigned C Carson Blair to the minor league camp.