Bauer (1-1) was recalled earlier in the day for his second start this season, and the right-hander held the AL's top hitting team to just two runs and seven hits in six-plus innings.
Mike Aviles hit a two-run double in the second when the Indians scored four runs off Verlander (5-3), who didn't settle in until the damage was already done.
David Murphy homered in the seventh for the last-place Indians, who have pulled two games closer to the Central-leading Tigers and can sweep the three-game series Wednesday.
Torii Hunter and Alex Avila homered for Detroit, which has lost two straight road games after winning 11 in a row.
The Indians were anxious to see how the head-strong 23-year-old Bauer, with the unorthodox warm-up routine and blazing fastball, would perform against a lineup featuring the likes of Miguel Cabrera, Victor Martinez and Hunter. Bauer has been dominating minor league hitters at Columbus, and had never faced any hitters of this caliber.
He was more than up for the challenge.
Obviously, it helped that he was staked to an early four-run lead and Detroit leadoff hitter Ian Kinsler ran his club out of a potentially big inning in the fifth.
Bauer held Hunter, Cabrera and Martinez to a combined 2 for 8, and he was helped by two double plays.
In the sixth, Indians manager Terry Francona won an instant-replay challenge of a close play at first. After initially ruling Detroit's Nick Castellanos safe at first, the umpires reviewed the video and overturned their call, resulting in an inning-ending double play.
When Bauer was lifted for Bryan Shaw after giving up a leadoff single in the seventh, he received a standing ovation from the Progressive Field crowd.
Shaw pitched two perfect innings and Cody Allen worked a 1-2-3 ninth.
The Indians grabbed a 5-1 lead in the second, when they collected three doubles, two singles and Michael Bourn caught Detroit napping and stole third.
The unexpected outburst began with a base-running blunder as Carlos Santana singled off Cabrera's glove at first but was thrown out trying to stretch it into a double. Nick Swisher doubled and Yan Gomes walked before Aviles lined his two-run double into the gap in left-center.
Bourn followed with a double off the wall in right to score Aviles and Cleveland's leadoff hitter alertly swiped third without a throw. Asdrubal Cabrera's RBI single made it 4-1 and brought Tigers pitching coach Jeff Jones out to visit Verlander, who dropped to 9-11 with a 5.36 ERA in his career at Cleveland.
Hunter gave the Tigers a 1-0 lead in the first with his sixth homer, a laser-like, 429-foot shot.
Lonnie Chisenhall hit a run-scoring single in Cleveland's first at-bat to tie it.