The two wins also improved Boston's record to a Major League-best 28-12 and gave the Red Sox a 9 1/2-game lead over the second-place New York Yankees, their biggest lead since September 1995.
Brendan Donnelly (2-1) relieved starter Curt Schilling to begin the seventh and earned the win.
Schilling labored through six innings, throwing 118 pitches (74 strikes), but he allowed just two runs.
After belting a solo homer in Tuesday night's win off Tim Wakefield, Brandon Inge again burned the Red Sox, this time with a solo homer on Schilling's first pitch of the fourth that just cleared the 37-foot Green Monster, giving the Tigers a 2-1 lead.
Schilling managed to scatter eight hits, all of them for extra bases -- seven doubles and a solo homer. It was a start similar to the one he produced last Aug. 10 at Kansas City, when the right-hander yielded nine doubles and a homer. He also walked an uncharacteristic four batters, matching his high in a Red Sox uniform.
"It was a monumental struggle and one of the weirdest games I have ever pitched in," Schilling said.
But Detroit could not capitalize, collecting just one hit in its first 16 chances with runners in scoring position. Schilling stranded 11 Tiger runners through his six innings.
Hinske, starting in right field for the injured J.D. Drew, helped Schilling with the catch of the night, diving for and catching a drive down the right-field line off the bat of Mike Rabelo.
"No, I don't even remember it," Hinske said of his up-close-and-personal meeting with the warning track. "I just remember looking up at my glove to make sure the ball was still in there, because usually it bounces out. I would have looked bad if that ball got by me, but I committed and was able to come up with it."
The spectacular catch, coming with runners on second and third and two outs, ended Detroit's fifth-inning rally. Hinske lay on his stomach for several moments after his chin snapped into the warning-track dirt.
"It was all over my mouth, eyes and nose," Hinske said. "The back of my neck is kind of sore right now. They told me I would have kind of whiplash symptoms tomorrow, but the adrenaline's going now, so I'm happy."
"He's always ready," manager Terry Francona said. "He just willed himself to catch that ball, and we got out there as quickly as we could. And he said, 'I'm OK. I just hit my head.' He hit it pretty hard. Doctors got down there right away, behind the dugout, and checked him out."
Most important, Hinske's heroics kept Detroit's lead at just one run. The Sox tied it an inning later, and Hinske was once again in the center of the action. He reached on an error by Inge and scored when Alex Cora beat out an infield single with two outs.
"Unbelievable," said a grateful Schilling. "One of the best catches I have ever seen anywhere."
Then came the bottom of the seventh. With the score tied at 2, Hinske drilled a fastball from Tigers reliever Wilfredo Ledezma (3-1) into the Boston bullpen with Jason Varitek aboard. A triumphant Hinske raised his fist as he rounded first base.
"Probably the home run, because it didn't hurt," Hinske said when asked which big moment felt better. "It just felt good being out there helping the team win. It was a big game for us. The catch, there was a guy on second base, and I was just trying to save a run there, and it turned out well. [And on the home run], I just got a good pitch out over the plate and put on good swing on it."
Before the homer, Hinske had been hitless in five previous at-bats covering the two games.
"He had a tough day at the plate going into that at-bat, but he willed himself to catch that ball in the outfield, and right before he came up, I told [bench coach Brad Mills], 'If there's any justice in this world ...' and he took a beautiful swing," Francona said. "To hit a ball that way for a home run, you've got to hit it pretty well."
Following the explanation of his great night to reporters, Hinske was treated to one more surprise. His two excited French bulldogs, Shady and Spike, came into the clubhouse to greet their master.
On this night, Hinske was the true master of his domain.