Baltimore scored a single run in four straight innings beginning with the third to overcome a 3-0 deficit, but the Royals sent 11 men to the plate in the sixth to take command.
Melky Cabrera and Billy Butler had back-to-back singles to drive in the first two runs of the inning. Navarro later had a two-run double and Getz delivered a two-run single.
Kansas City produced eight of its 17 hits in the sixth.
Although the Royals are 15 games below .500 for the season, they have a winning record (12-11) against teams from the American League East.
Blake Wood (5-0) earned the victory with two scoreless innings in relief of starter Jeff Francis.
The loss was charged to Zach Britton (6-9), who gave up four earned runs on 12 hits in 5 1/3 innings.
In his previous start Britton retired only one batter while surrendering seven runs to the New York Yankees.