Detroit twice battled back from late deficits, scoring two runs in the seventh to tie the game at 3-3 and a solo home run from J.D. Martinez in the ninth inning to force the game into extras.
Brantley's home run in the 10th gave Cleveland its first win in five days and snapped a streak of 11-straight road wins for Detroit.
Justin Verlander will look to put the Tigers back on track Tuesday, facing a team that he has seen 36 times in his career - more than any other team.
TV: 7:05 p.m. ET, FSN (Detroit), SportsTime Ohio (Cleveland)
Verlander has dominated Cleveland hitters in his 10 Major League seasons, having surrendered just 203 hits compared to 256 strikeouts. Nine players on the Indians' roster have at least 20 career at-bats against the former MVP, and six of them own averages lower than .220 in those situations.
Monday night's hero owns the best career numbers against Verlander: 16 hits, eight RBI and a .390 batting average. Brantley has two extra-base hits against Verlander in 41 at-bats, but has never hit a home run off of him.
Cleveland will counter with its No. 3 prospect Trevor Bauer, who makes just his second MLB start of the season. Bauer is replacing struggling Indian Danny Salazar in the starting rotation after pitching in the minor leagues since his only start of the season on April 9. The 23-year-old was handed a tough-luck loss in that game after surrendering just one earned run in six innings.
The Tigers have never faced Bauer, whose control is the only problem keeping him from becoming a quality Major League pitcher. Bauer features an elite fastball and electric curveball that place him among the top pitching prospects in the American League. He will also throw a changeup and slider when he's ahead in the count.
Despite the loss Monday, Detroit stands 9.5 games ahead of Cleveland and maintained its seven-game lead in the AL Central.
The Tigers will finish the three-game series with Cleveland on Wednesday afternoon before heading back home to Comerica Park.