Milone (8-7) limited the Royals to two hits through eight innings before giving up two doubles and two singles in succession with one away in the ninth.
Grant Balfour came on with one out and two on and eventually recorded his 22nd save.
Crisp produced a sacrifice fly in the fifth that gave the Athletics a 3-0 lead and he singled home two more runs during Oakland's three-run ninth inning.
Those insurance runs turned out to be vital when the Royals tried to put together a rally in the bottom half of the inning.
Oakland opened a six-game road trip that will also take the Athletics to Pittsburgh for three contests between two of baseball's top teams.