Ortiz, a former Twin, hit a two-run homer in the sixth and finished a triple shy of the cycle. Jarrod Saltalamacchia added a homer and two RBIs to help the Red Sox win for the fifth time in seven games and move 1½ games ahead of the idle Yankees in the AL East.
Jason Kubel had a homer among his three hits and two RBIs for the Twins, who have lost five straight.
Alfredo Aceves (8-1) gave up one run in an inning in relief of Tim Wakefield for the victory. Wakefield gave up five runs — three earned — and eight hits with five strikeouts. But he missed out on victory No. 200 for the third straight start when Aceves gave up the game-tying single to Kubel in the eighth.
Glen Perkins (3-2) gave up two runs — one earned — in 1 2-3 innings to take the loss. He left with the game tied and a runner on second, but Joe Nathan gave up a single to Ortiz and an RBI-double to Saltalamacchia to let the game slip away.
Scott Baker gave up five runs and nine hits with five strikeouts in six innings for the Twins.
The Twins scored just four runs — and committed five errors — in a weekend sweep at the hands of the Chicago White Sox, a series so ugly that manager Ron Gardenhire said his team "didn't play worth a flip."
They showed more fight on Monday, but it wasn't enough against the best team in the AL.
After a four-hour, 15-minute grinder against the Yankees on Sunday night, the Red Sox didn't arrive at their Minneapolis hotel until just before 5 a.m. Their legs looked a little heavy and their bats a little slow in the early going as they tried to shake the cobwebs for yet another nationally televised game.
The Twins scored three times in the second and Kubel led off the fourth with a homer to give the Twins a 4-1 lead.
A considerable contingent of Red Sox fans started with the chants of "Let's go Red Sox!" in the sixth, and it appeared to wake up their team. Carl Crawford led off with a triple and scored on a sacrifice fly from Adrian Gonzalez. Ortiz hit a two-run homer and Saltalamacchia made it back-to-back with a solo shot that tied it 5-5.
Jed Lowrie had one hit in his first game since June 16, when he went on the disabled list with a strained left shoulder.
NOTES: Twins SS Tsuyoshi Nishioka had the day off to be with his wife for the birth of the couple's first child. Baby girl Sena was born at 8:26 a.m. and weighed in at 6 pounds, 9 ounces. ... Twins GM Bill Smith said minor league RHPs Kyle Gibson and Anthony Slama met with team doctors to discuss their elbow injuries. Both will undergo further tests on Tuesday before a determination is made. ... Boston manager Terry Francona gave 3B Kevin Youkilis the day off. 2B Dustin Pedroia hit in the cleanup spot. The Red Sox were 3-0 in his only other three games there this season. ... The homers by Ortiz and Saltalamacchia were the sixth back-to-back set for the Red Sox this season. ... LHP Francisco Liriano (7-9, 5.03 ERA) will take the mound for the Twins in Game 2 on Tuesday against LHP Erik Bedard (4-7, 3.55). ... It will be Bedard's second start for the Red Sox since coming in a trade from Seattle. He gave up three runs and seven hits with five strikeouts in five innings of a no-decision against Cleveland on Aug. 4. ... Liriano has been wildly inconsistent all season and is coming off one of his worst starts of the year when he gave up seven runs and 10 hits in five innings in a loss to the Angels.