Michael Tucker's RBI single was followed by a double by Joe Randa as Kansas City built a three-run cushion. Raul Ibanez and Brent Mayne added sacrifice flies to stretch the Royals' lead to 6-1.
Byrd (2-0) gave up two runs and five hits with three strikes and no walks. He is Kansas City's only pitcher with more than one win this season.
Cory Bailey bailed out Byrd in the eighth, striking out Nomar Garciaparra with runners at first and second and two out.
Bailey fanned two more in the ninth for his first career save.
Lowe, who flirted with a no-hitter on Friday in Baltimore, was the unlucky loser after yielding two runs and eight hits in seven innings. He struck out three and walked one.