Rusin (1-0), called up from Class AAA Iowa before the game after scheduled starter Garza was dealt to the Texas Rangers, limited the Diamondbacks to two runs on two hits.
The 26-year-old left-hander walked two and struck out four as the Cubs took the first of a four-game series.
Junior Lake went 4-for-5 with a two-run homer and three RBI and Dioner Navarro also went deep, sparking Chicago's offense.
Tyler Skaggs (2-2) allowed three runs on seven hits over six full frames as the Diamondbacks fell out of first place, dropping one-half game behind the Los Angeles Dodgers in the National League West with the loss.
Paul Goldschmidt went 2-for-4 with an RBI and Martin Prado also drove in a run for Arizona.