Kennedy (1-0) surrendered a first-inning run but then clamped down to hold the Cardinals scoreless over the next five frames as the Diamondbacks scored four times to take control.
The right-hander ended up allowing two runs on five hits, walking one and striking out eight.
Pollock followed with another two-bagger, scoring Kubel and Goldschmidt to put the D-backs ahead to stay.
Adam Wainwright (0-1) was tagged for four runs (three earned) on 11 hits over six full frames for the Cardinals.
Matt Holliday and David Descalso each drove in a run for St. Louis.
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