Garza (5-7) limited the D-backs to five hits and a walk, and struck out seven for the Cubs, who held Arizona to a total of three runs during the weekend series at Wrigley Field.
Darwin Barney socked a two-run homer in the second inning and Anthony Rizzo added a run-scoring single for the winners.
Arizona's Trevor Cahill (7-8) was tagged with the loss, yielding three runs on four hits in a 5 1/3-inning stint.
Aaron Hill went 2-for-3 with a solo homer for Arizona.
Celebrity Families of 2014 [PHOTOS]