Choo finished 2-for-4 with two runs scored for the Reds, who took three of four from the Diamondbacks.
Aaron Hill followed with a hard-hit grounder to third that went off Zack Cozart for a base hit to score Eaton.
Cahill settled down after his shaky third inning, allowing just two hits the rest of the way until he was pulled for a pinch-hitter in the eighth.
The right-hander was charged with one run on four hits, striking out seven.
His four walks all came in the first three innings.
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Paul Goldschmidt had three hits for Arizona, which fell to 3-4 on its current 10-game road trip.