Bobby Seay (2-2) worked a scoreless 11th for the win. Miner then gave up a run and two hits in the 12th before finishing.
"Other than the hits and run, it was a lot of fun," Miner said, jokingly adding he's hoping to become the career saves leader someday. "Hoffman better watch out."
Hoffman entered the Milwaukee Brewers' game at San Diego on Saturday night with 577 career saves. Miner had been 0 for 7 in save chances before converting.
Jose Veras (4-2) walked Placido Polanco and Clete Thomas to open the Tigers 12th before striking out Miguel Cabrera and getting Carlos Guillen to foul out to third. Raburn, who came in batting .167 in his last 13 games, lined a 1-1 pitch to right, scoring Polanco. Veras then balked home Thomas to make it 4-2.
"I just didn't make the right pitches," said Veras, who had a 1-2 count on both men he walked. "You walk guys, you are in trouble."
It was a difficult end to a trying week for the Indians, who fell 18 games under .500 with the loss on Victor Martinez bobblehead doll night. Martinez was traded to the Boston Red Sox on Friday in the latest deal for the rebuilding Tribe.
"We tied it in the ninth and created some chances later to win it," Indians manager Eric Wedge said. "I can't say enough about the effort."
Jamey Carroll had a one-out single off Miner and Trevor Crowe followed with a double to the gap in left-center. Carroll scored on a groundout by Wyatt Toregas and Miner got Grady Sizemore to pop out on the next pitch to end it.
"The runner on third didn't mean anything and I got a ground ball there," Miner said. "There's probably a lot of guys I'd rather face with the game on the line than Grady, but fortunately I got him out."
The AL Central-leading Tigers' second win in six games kept them 1 1/2 games ahead of the Chicago White Sox. Detroit lost 6-5 to Cleveland in 13 innings Friday night.
A crowd of 31,353 showed up to get their Martinez bobbleheads and watch Sandy Alomar Jr. go into the Indians' Hall of Fame during a pregame ceremony. Former owners Bill Veeck and Richard Jacobs and right-hander Wes Ferrell were inducted posthumously.
It was a surprising turnout considering the negative fan reaction stemming from a series of trades this week that sent away first baseman Ryan Garko, reigning AL Cy Young Award winner Cliff Lee and three-time All-Star catcher Martinez.
Rick Porcello allowed one run and four hits over eight innings against Cleveland's depleted lineup. He came in with a 1/3 record and 9.16 ERA in his last four outings.
Peralta tied it at 2 with a broken-bat sacrifice fly in the ninth against Fernando Rodney, who blew a save for the first time in 22 chances this season.
Sowers allowed two runs and four hits over six innings. He was replaced by right-hander Justin Masterson, acquired in the Martinez deal. Masterson fanned the first batter he faced, Everett, and allowed two hits over three scoreless innings.
The teams wore replica jerseys of Negro League franchises that played in the two cities 70 years ago. ... Toregas lined a single to left in his first career at-bat in the third inning. "I was shaking so hard I almost fell down when I went to the plate," Toregas said. "I hit the ball, but have no idea how I did it." ... Leyland said RHP Jeremy Bonderman (shoulder) and LHP Nate Robertson (elbow) are expected to begin minor league rehabilitation assignments soon. Both threw simulated games Friday. ... A scoring change from Friday's game took an error from Polanco and gave it to CF Curtis Granderson, whose low throw skipped away from the infielder and allowed a runner to advance one base.