Bauer struck out Judge three times, twice getting the MVP candidate looking. He allowed just two hits in 6 2-3 innings before manager Terry Francona turned to baseball's best bullpen, using Andrew Miller and closer Cody Allen to finish the three-hitter.
Allen came in with two on and two outs in the eighth to face Judge, who struck out for the fourth time and angrily snatched at his bat frustration.
Bruce connected for a two-run homer in the fourth off Sonny Gray and added a sacrifice fly in the fifth as the Indians began a journey to try and end the majors' longest Series title drought.
Corey Kluber, an 18-game winner during the regular season, will start for the Indians in Game 2 on Friday against CC Sabathia.