Bauer (1-1), the Diamondbacks' third overall pick from UCLA in last year's draft, hamstrung the Dodgers while making only this third major league start.
The right-hander yielded only two hits and one walk, and struck out six while retiring the final 12 Los Angeles batters he faced.
Patrick Corbin finished with three innings of one-run relief to notch his first career save for Arizona, which took three wins in its four-game series with the Dodgers.
Paul Goldschmidt went 2-for-3 with a three-run homer and four RBI, while Justin Upton added three hits, including a run-scoring triple, and scored three times in the victory.
Dodgers starter Chris Capuano (9-4) was tagged for five runs on nine hits in a five-inning effort.
Juan Uribe got the only RBI for first-place Los Angeles, which dropped its third straight but still hung onto a half-game lead over San Francisco in the National League West entering the All-Star Game break.