Jackson worked out of further trouble and Cabrera hit a go-ahead single in the eighth inning to help the Tigers win their sixth straight game, 4-3 over the Rangers.
The Rangers arrived in Detroit this week with a seven-game winning streak but managed only seven runs in their three losses.
"We just got beat," manager Ron Washington said. "In the last two games, we were one hit away from the game being different.
"Today, we were one hit away from the game being different on three different occasions. We just couldn't get the knock," he said.
Jackson (4-2) won his third straight start, throwing a career-high 132 pitches in eight innings.
Manager Jim Leyland visited the mound before the Rangers tied the game but said he had no intentions of pulling Jackson, even during the rocky eighth.
"He's a warrior," Leyland said. "I felt like he deserved to be out there."
"It would've been easy to go to the 'pen," Jackson said of Leyland's decision to stick with him, "but I'm glad he had that confidence. It rubs off on a pitcher."
Jackson gave up seven hits, walked five and struck out seven. He has a 1.64 ERA and 21 strikeouts in his last three starts.
With the score 3-all, Millwood retired the first two batters in the eighth. Placido Polanco singled, Thomas walked on a full count and Cabrera singled.
David Murphy hit a solo homer in the Texas first. Thomas tied it in the bottom half with his first home run of the season.
The Rangers loaded the bases with one out in the second, but Elvis Andrus flied out to shallow right and Thomas made a perfect throw to catch Cruz at the plate.
Millwood walked the leadoff man in the third and gave up Granderson's 11th homer of the season a batter later.
Detroit sent outfielder Wilkin Ramirez back to Triple-A Toledo after the game to make room for right fielder Magglio Ordonez (bereavement list), who will join the club for Friday's game against Colorado. ... Texas' five stolen bases were its most since Aug. 6, 2001. ... Rangers center fielder Josh Hamilton (groin) missed his third straight start but Washington said he hoped to have the All-Star and closer Frank Francisco (shoulder) back when Texas plays Friday night in Houston. ... Leyland said outfielder Marcus Thames (rib cage strain) should begin a rehab assignment at Toledo next week.