McCutchen, who also provided an RBI double in the first, drove in Starling Marte with a sacrifice to break a 3-3 tie and Garrett Jones added a solo homer in the eighth for insurance.
Pirates starter Wandy Rodriguez came out of the game with hamstring tightness in the third inning after yielding two runs (one earned), and Jeanmar Gomez (1-0) relieved, earning the win with 1 2/3 frames.
Arizona's Trevor Cahill (0-2) absorbed the loss, allowing four runs on four hits over five full innings.
Paul Goldschmidt, Alfredo Marte and Wil Nieves each drove in a run for the D-backs.
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