Doubront allowed two runs, one earned, striking out five and walking three, while the American League East-leading Sox got homers from Jacoby Ellsbury and Jonny Gomes in their first outing after the All-Star break.
The win kept them 2 1/2 games clear of the Tampa Bay Rays, who beat the Toronto Blue Jays 8-5 in a home run fest at the Rogers Center.
Ben Zobrist, Wil Myers, Evan Longoria and Kelly Johnson all went deep for the Rays as they rallied from 4-1 behind for the win.
Jose Bautista's two-run shot was one of four long balls for the Jays.
Brad Miller hit his first two Major League home runs to lift the Seattle Mariners to a 10-7 win over the Houston Astros.
The rookie shortstop's offensive outburst - he also drove in five runs - meant Brandon Barnes ended up on the losing side despite hitting for the cycle.
Kyle Seager and Justin Smoak also went yard for the Mariners.
Ervin Santana hurled 7 1/3 scoreless frames to lift the Kansas City Royals to a 1-0 win over the Detroit Tigers, ending a five-game losing streak.
Santana allowed two hits, fanning six and walking one, while Billy Butler's RBI single in the first was the only offence either side could produce.
Jake Westbrook added three hits, two runs and an RBI to his 6 1/3 innings of two-run ball as the St Louis Cardinals beat the San Diego Padres 9-6.
All-Star Brandon Phillips' bases-clearing double lifted the Cincinnati Reds to a 5-3 win over their National League Central-rival Pittsburgh Pirates.
Matt Wieters homered as manager Buck Showalter earned his 250th win in the Baltimore Orioles' 3-1 win over the Texas Rangers.
Andre Ethier launched a game-winning homer in the top of the ninth as the Los Angeles Dodgers beat the Washington Nationals 3-2.
Michael Young and Chase Utley both homered and drove in three as the Philadelphia Phillies beat the New York Mets 13-8.
Solo shots from Juan Francisco and Carlos Gomez backed six shut-out innings from Kyle Lohse in the Milwaukee Brewers' 2-0 win over the Miami Marlins.
Albert Pujols, Erick Aybar and Mike Trout all went deep in the Los Angeles Angels' 4-1 win over the Oakland Athletics, while the San Francisco Giants edged out the Arizona Diamondbacks 2-0.
The Minnesota Twins beat the Cleveland Indians 3-2, the Atlanta Braves were 6-4 winners over the Chicago White Sox and the Chicago Cubs overcame the Colorado Rockies 3-1.