Santiago drew a bases-loaded walk from Ramon Ramirez with two outs in the 12th inning, then started toward the dugout before his teammates reminded him that he had to go touch first base.
"In all of my career, I've never been a part of anything like that, but the guys were all telling me to go," Santiago said. "It's one of those things you learn, I guess."
The Tigers were down 6-1 after five innings and 6-4 going into the bottom of the eighth. Still, they won for the sixth time this season when trailing after six.
"It's a great win," manager Jim Leyland said. "Obviously, it didn't look good early, because we put ourselves behind the 8-ball, but we hung in there and got ourselves a nice, exciting win."
After homering in the eighth, Magglio Ordonez opened the 12th with a single off Manny Delcarmen (1-2) and went to third on a one-out single by Brennan Boesch off Scott Schoeneweis -- his fourth hit of the game.
Jose Valverde (1-1) got the victory with an inning of scoreless relief.
"It's always great to get a win like that, but especially against a very competitive team like Boston," Inge said. "Coming back against them is a very good sign."
The Red Sox had runners in scoring position in both the 10th and 11th, but Joel Zumaya got out of both jams.
"There's no way we should have lost that game," said Kevin Youkilis, who finished with a career-high five walks. "The good thing is that we've got another game in 13 hours, so we won't have time to dwell on it."
Neither starter got a decision. Jon Lester allowed four runs on four hits and four walks in seven innings. He struck out 10.
Detroit's late rally saved Dontrelle Willis from a loss after he gave up four runs, four hits and seven walks in 3 1/3 innings.
Willis struck out two of Boston's first four hitters in the third, but walked the other two. David Ortiz broke the scoreless tie with an RBI single, and after Willis walked Adrian Beltre to load the bases, J.D. Drew made it 3-0 with a ground-rule double to center.
He left with two on in the fourth and Ortiz increased Boston's lead to four runs with a two-out RBI single off Eddie Bonine.
Still, given Willis' performance, the Red Sox knew they should have been ahead by even more.
"At the time, we didn't think that was going to cost us the game, but we definitely missed some chances in the early innings," manager Terry Francona said. "It turned out to hurt us more than we thought."
Bill Hall made it 6-1 in the fifth with his second homer in two days, a two-run shot off Bonine.
Boesch's two-run triple pulled the Tigers to 6-3 in the sixth. Inge made it a two-run game with a sacrifice fly.
"I had really gotten into a rhythm, but that sixth inning hurt me," Lester said. "I only really made one bad pitch, but Boesch hit it hard."
Detroit LF Casper Wells made his major league debut, batting second. Wells, called up Friday, was scheduled to return to Triple-A Toledo on Sunday to make room for RHP Armando Galarraga, but those plans changed after the game. Instead, the Tigers promoted Galarraga and infielder Danny Worth, and demoted Sizemore and RHP Max Scherzer. ... Beltre is 8 for 10 in his career against Willis. ... Tigers reliever Zach Miner sustained a partial tear of an elbow ligament while pitching in extended spring training and is expected to miss the rest of the season. ... The game was delayed for a few moments in the top of the ninth after plate umpire Ron Kulpa was hit in the mask by Jeremy Hermida's foul ball on a 100 mph fastball from Zumaya. Kulpa stayed in the game after being examined by Tigers trainer Kevin Rand.