The Orioles won for just the second time in eight games.
Eaton (1-2) enjoyed a 6-0 lead heading into the eighth, but put two runners on with one out before being relieved by Jim Johnson. Both runners scored on Jim Thome's single, leaving Eaton with a two-run, six-hit effort over 7 1/3 innings.
Luke Scott doubled home two runs in Baltimore's three-run third, boosting the Orioles' lead to 5-0.
Bartolo Colon (1-1) took the loss, giving up five runs on eight hits in just three innings of work.
Chicago wrapped up what had been scheduled as a 10-game road trip. One of those contests was rained out in Detroit.