Burton allowed two straight hits with two outs to set the stage for Drew's game-winning double that went just over the glove of Ryan Doumit in left field. It gave the Red Sox the win just before midnight ET at Fenway Park.
"We had some opportunities and they had some, but they got the big hit at the end," Twins manager Ron Gardenhire said. "It was a tough one for us. We threw just about everything we had at them."
The rally for the Red Sox was keyed by a two-out infield single from catcher Jarrod Saltalamacchia, as Burton didn't set his feet on the throw to first base, which pulled Justin Morneau off the bag.
"He just kind of rushed the throw," Gardenhire said. "If he sets his feet, he probably gets him."
It gave Drew the chance to be the hero and he delivered on a night where he went 4-for-5 with a homer and three RBIs.
"He's been swinging the bat, as we said, much better of late," Red Sox manager John Farrell said. "A big night for him, a key hit, obviously to walk it off."
It came after Minnesota tied it up with a solo shot from Brian Dozier with one out in the ninth off Boston closer Joel Hanrahan. Dozier's first blast of the year came on the heels of reliever Casey Fien giving up the game-tying homer in the seventh and the go-ahead homer in the eighth.
Fien entered the eighth with the Twins clinging to a one-run lead, but gave up a solo shot to Drew to right field to tie the game. Fien was able to get through the seventh, but came back out for the eighth and promptly allowed a solo homer to Dustin Pedroia with nobody out as Boston took a 5-4 lead.
"We had a lead and they had a couple big home runs on us," Gardenhire said. "We got a couple balls up and Drew got one off of Fien and then Pedroia had a heck of an at-bat and he got one. And we come back and tie it up and it goes from there.
It handed Vance Worley the no-decision, as he remains winless this season despite leaving in the sixth with a three-run lead.
The Twins jumped out to an early lead with two runs in the first against Red Sox right-hander Clay Buchholz. Joe Mauer and Josh Willingham hit back-to-back doubles off the Green Monster in left before Morneau drove home Willingham with a single up the middle. But the Twins left the bases loaded, as Oswaldo Arcia and Aaron Hicks struck out swinging.
Minnesota added a run in the fourth, when Hicks roped a first-pitch RBI double down the right-field line to score Arcia, who led off the inning with a double.
The Red Sox didn't score their first run until the bottom of the fourth on a solo shot from Shane Victorino off Worley.
It was more back-and-forth in the fifth, as both teams tacked on a run. A sacrifice fly by Morneau scored Mauer, who led off with a double and reached third on a single from Willingham. The Red Sox scored on a two-out RBI single from Drew. But Drew was later called out at home while trying to score from first on a double by Jacoby Ellsbury to end the inning, though replays showed Mauer never applied the tag.
The Red Sox chased Worley in the sixth, as Victorino and Pedroia started the inning with consecutive singles. Left-hander Brian Duensing came in and was able to get David Ortiz to ground into a double play, but gave up an RBI single to Mike Napoli.
With the inherited run scoring, Worley finished up allowing three runs on nine hits with three strikeouts over five-plus innings.
"I thought I had better stuff today," Worley said. "I had command of both sides of the plate. I was working down, but the last couple innings, I started to elevate a little bit. But for the most part, I'm pretty happy with the outing. But it would've been nice to get a 'W.'"