Gregorius then sent a single to shallow right field, scoring Ross to make it 4-4, and pinch-hitter Eric Chavez coaxed another base-on-balls, forcing in Montero with the decisive score.
Gregorius also went deep in the sixth for the National League West-leading Diamondbacks, who took a second straight from Milwaukee.
Will Harris (2-0) got credit for the victory and Brad Ziegler notched his third save for Arizona.
Lohse (5-7) was touched for five runs on four hits over 6 1/3 innings as Milwaukee dropped a third straight overall.
Jonathan Lucroy went 2-for-4 with a two-run homer for the Brewers.
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