Tyler Thornburg (1-0) earned the win with a perfect 10th inning. Burke Badenhop (0-1) came in to start the 11th and gave up Davis' one-out double, then Schafer one-hopped the Green Monster between the left and center fielders.
Francisco Rodriguez struck out the side in the 11th for his second save.
Jean Segura and Aramis Ramirez each had three of Milwaukee's 19 hits. Carlos Gomez and Mark Reynolds each hit a solo homer in the second inning - the first hit of the season for Reynolds, who was signed as a free agent on Jan. 17.
On a 40-degree night with the wind blowing the brand new World Series championship flag stiffly from left field to right, Milwaukee led 3-0 and 6-2 before Boston came back on Mike Napoli's three-run homer in the third and an unearned run in the sixth. The Brewers threatened in the eighth and again in the 10th, but both times Jonathan Lucroy lined out to Daniel Nava - at two different positions.
The Brewers scored one in the first, two in the second and three in the third off Clay Buchholz, who gave up a career-high 13 hits and lasted only 4 1-3 innings. But Boston made it 6-5 on Napoli's homer, then tied it in the sixth when Xander Bogaerts led off with a double and scored from third when shortstop Jean Segura fumbled Jonathan Herrera's slow roller for an error.
It was the second error of the game for the Brewers, leading to four unearned runs. Ramirez had an error at third base on Dustin Pedroia's grounder that kept the third inning alive; one out later, Napoli homered to deep center.
After scoring six runs in the first three innings, Milwaukee led off the fourth with back-to-back singles from Gomez and Segura. But Buchholz struck out Lucroy before Ramirez hit a long fly ball to center field.
Gomez tagged up and took third base and then tried to score, when second baseman Dustin Pedroia bobbled the throw from center fielder Grady Sizemore. But Gomez was out at the plate to complete the inning-ending double play.
The rally rescued Buchholz, who was 12-1 in an injury-plagued 2013 season, from matching his loss total from all of last year.
The Brewers put the go-ahead run on third in the eighth inning against Brandon Workman when Segura singled with two outs, stole second and took third when A.J. Pierzynski's throw went into center field. Lucroy lined the ball toward right field, but first baseman Daniel Nava leaped to snare it and end the inning.
Milwaukee also loaded the bases with two out in the 10th when Segura struck out but reached on a wild pitch. Lucroy hit one deep to right, but Nava had moved to the outfield and he went back to get that one, too.
Notes: Ryan Braun was out of the lineup, one day after going 0-for-5 on Friday in the series opener to extend his slump to 14 hitless at bats. He is 1-for-16 on the season. ... World Series MVP David Ortiz was not in the starting lineup for the Red Sox for the first time of the season. He pinch hit for Jackie Bradley Jr. in the ninth, but grounded out. ... The Red Sox shuffled their lineup repeatedly before the game, scratching Will Middlebrooks (right calf) and Mike Carp (lower back tightness). ... Red Sox closer Koji Uehara struck out all three batters he faced in the top of the ninth.