The Monday nighter at Fenway Park took 4:44 to complete, ending with Stephen Drew's two-out double off the left field wall in the bottom of the 11th that plated Jared Saltalamacchia with the winning run.
Drew's game-winner capped a 4-for-5 night at the plate for the Boston shortstop. Two of his other hits also produced runs. In the fifth, he singled in a run to get the Red Sox down by two, 4-2, and his seventh inning solo homer tied the game at 4-4. Dustin Pedroia had a solo shot in the eighth to put the Sox ahead by a run, but the Twins' Brian Dozier answered with a blast of his own to re-tie the game at five in the ninth.
The win left the Red Sox two games up on the idle Orioles and Yankees in the AL East and dropped Minnesota five games behind Detroit in the Central.
Game two of the four-game series is slated for 7:10 pm EDT, Tuesday night.
The Braves split a four-game series with the Nationals and won one of two with the Mets before opening their three-game set in Cincinnati, their second straight win, spearheaded by shortstop Andrelton Simmons.
Simmons, who batted .279 in 49 games for the Braves in 2012, still qualifies as a rookie, his 3-for-4 effort one of his best of the season. He provided most of the mojo in the victory, socking a solo shot in the second inning to put the Braves up 2-0, singling in the third to drive in another run and putting the game away with a two-run blast in the eighth to Atlanta up, 6-2.
The two homers gave Simmons four on the season, with 15 RBI.
The loss sent the Reds into a second place tie with Pittsburgh, both trailing the Cardinals - winners of six straight - by three games.
The Braves and Reds meet again on Tuesday at 7:10 pm EDT.