The decisive hit capped a comeback in which the Rockies forged a 5-5 tie in the bottom of the ninth on Dexter Fowler's two-out, two-run double.
Justin Upton's throw was offline, and Rosario crossed the plate with the winning run.
Belisle (1-1) earned the victory with two scoreless frames.
Jonathan Herrera went 3-for-4 with a run scored for Colorado, which broke a three-game losing streak.
Hudson allowed three runs on six hits over six innings in the no-decision.
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