Arizona had grabbed a 6-5 lead in the eighth on a Wil Nieves solo homer, but the Orioles battled back to tie the contest in the top of the ninth on Nick Markakis' sacrifice fly off Brad Ziegler (6-1).
It didn't take long for the D-backs to respond, however, as Eaton (two hits, two RBI) drilled the first pitch he saw from Darren O'Day (5-2) leading off the bottom of the inning over the right-field fence for a game-winning clout, his first homer of the year.
Aaron Hill went 2-for-4 with a solo blast and two RBI as Arizona notched a victory in the first of a three-game series.
Chris Davis belted his MLB-best 43rd homer of the year, Matt Wieters also went deep and Brian Roberts drove in a pair of runs for Baltimore, which trails Oakland by 2 1/2 games for the final American League playoff spot.
Notable deaths of 2014 [PHOTOS]