The Tigers led 5-0 and Justin Verlander didn't allow a hit until the sixth inning, but new closer Francisco Rodriguez gave up three runs in the ninth, including two-out RBI doubles to pinch-hitter Derek Dietrich and Dee Gordon.
The Tigers hit three homers but went 1 for 11 with runners in scoring position before Kinsler put them ahead in the 11th. Detroit also had two runners picked off first base in the final three innings.
Giancarlo Stanton homered for the Marlins.
Anthony Gose walked to start the 11th against Craig Breslow and went to second on a sacrifice. Kinsler followed with a one-out RBI single.
Kinsler hit a three-run homer in the second inning and scored twice. Gose and pinch-hitter Victor Martinez hit back-to-back homers in the ninth off Bryan Morris.
Drew VerHagen (1-0) escaped a two-on, two-out jam in the 10th. Shane Greene pitched a perfect 11th for his first career save.
Rodriguez was unable to nail down the victory in the ninth. With the Marlins trailing 7-4, Martin Prado singled to lead off and J.T. Realmuto hit a one-out double.
After left fielder Justin Upton made a running catch on the warning track to rob Adeiny Hechavarria for the second out, Dietrich and Gordon doubled. Marcell Ozuna then popped out to send the game into extra innings.
Rodriguez was 38 for 40 in save chances with the Brewers last year.
Verlander blanked Miami for five innings and had a broken-bat single to become the first American League pitcher to get a hit on opening day since Milwaukee's Bill Parsons and Oakland's Ken Holtzman in 1972, according to STATS.
Stanton, who missed the second half of last season with a broken hand, hit a two-run shot off Verlander in the sixth. Gordon, the 2015 NL batting champion, broke up Verlander's no-hit bid with a double and finished with three hits.
The game marked the debut of a cozier Marlins Park, with the fence moved in during the offseason, and the changes might have made the difference on Martinez's homer to center field.
Detroit and Miami were the last clubs to begin their season, and the home team didn't seem ready.
Marlins newcomer Wei-Yin Chen, the most significant offseason addition to their roster, had a rough time in his first opening day start. He allowed five runs and nine hits in five innings.
In the second inning, Chen was hit on the left arm by a line drive off Gose's bat. Verlander followed with a single for his third career hit, and Kinsler homered to make it 4-0.
The game was announced as a sellout, with attendance of 36,911. But there were several thousand empty seats -- and a significant number of Tigers fans who were chanting for their team in the first inning.
TRAINER'S ROOM
Marlins LHP Mike Dunn pitched on opening day in each of his seven previous major league seasons, but he was on the disabled list Tuesday with a strained left forearm. He said he planned to begin playing catch Wednesday and hopes to be activated for the start of the next homestand April 15.
UP NEXT
On Wednesday, Miami will start ace Jose Fernandez, who in his final Marlins Park outing last year became the first pitcher in the modern era to win his first 17 career home decisions.
Anibal Sanchez will start for Detroit against his former team. He went 44-45 for the Marlins in 2006-12.
