Millwood threw seven shutout innings before handing a 4-0 lead over to C.J. Wilson, who failed to make it out of the eighth.
"Millwood was doing a great job of shutting us down, so we were glad to get to their bullpen," Inge said. "Our pitchers kept us in the game against some massive hitters, which is the key, because we've got a great offense."
Brandon Lyon (1-1) threw two perfect innings and Fernando Rodney got three outs for his second save in as many days. Detroit relievers have pitched seven innings in the last two games without allowing a baserunner.
"We've still got some bridging to do with our bullpen, but they have been tremendous in the last two days when we've needed them," manager Jim Leyland said. "That's their job -- don't let teams add on runs, so our offense has a chance at the end."
"I was able to get guys out when I needed them, but I threw too many pitches," Millwood said. "This is frustrating, because we start with a sweep at home and then get swept here."
Wilson (0-1) entered to pitch the eighth and Inge greeted him with his 100th career homer.
"I'd been pulling away from Millwood's cutter, so I wanted to stay on the plate and hit something hard to right," he said. "I wasn't thinking about a homer, but I'll take it."
After Adam Everett singled, Curtis Granderson hit a sinking liner that went under rookie shortstop Elvis Andrus' glove for an error. That put runners on first and second for Polanco, who chopped an RBI double over third base.
Wilson threw a called third strike past Magglio Ordonez for the first out and intentionally walked Cabrera to load the bases. Guillen followed with a double to tie it at 4.
"No one wants to face Cabrera the way he is hitting right now," Guillen said. "I'd walk him to pitch to me, but I had a good at-bat and got a good swing on a slider."
"I just couldn't stop the bleeding," Wilson said. "They got some broken-bat hits and some solid ones, but you don't always get those breaks."
Hamilton hit his first homer in the first inning and Kinsler connected for his second of the season in the third.
The Rangers put two runners on in the fourth and fifth, but Edwin Jackson got out of both jams. Jackson allowed five hits in six innings.
"I made the two mistakes and got us in a hole, and then it was a matter of damage control," he said. "I know how good our offense is, so I just wanted to keep it close."
RHP Joel Zumaya (shoulder) threw one inning in a rehab appearance for Class A Lakeland. He walked the first two batters he faced and allowed two runs -- one earned -- but Leyland said he was happy with the reports of Zumaya's velocity. ... Detroit's Jeremy Bonderman (blood clot) is expected to throw off a mound on Monday. ... Guillen left the game after his eighth-inning double with recurring soreness in his right Achilles' tendon, but said he expects to be fine. ... Inge has hit 56 of his 100 homers at Comerica Park, the most of any player since the stadium opened in 2000, and needs one game to become the 14th active player to have 1,000 career games with the same team.