Hinske's blast off John Axford (0-1) over the center-field wall put the Diamondbacks on top by two runs.
The Brewers responded with a run in the next half-inning, but the D-backs limited the damage and completed their first-ever three game sweep of Milwaukee.
Tony Sipp (1-0) earned with decision with a 1-2-3 10th inning.
Paul Goldschmidt went 3-for-5 with two RBI and Aaron Hill bashed a two-run homer for the winners.
Norichika Aoki went 4-for-6 with three RBI for Milwaukee, losers of five straight.
'Happy' fan videos make Pharrell cry [VIDEO]
Pregnant Mila Kunis wins 'Best Villain' at MTV Movie Awards