Doubront (8-5) gave up five singles, struck out five, didn't walk a batter and allowed just one runner to reach second base in holding an opponent to three runs or fewer for the 15th straight start. It's the most by a Red Sox left-hander since at least 1920.
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"Being consistent in my five-day routines and just taking my plan for my next outing, that's the most important one to me," he said. "We're working on that every fifth day."
The 25-year old left-hander's current string of three-run or fewer games is the best since by a Boston pitcher since Tim Wakefield went 17 straight in 1995.
"Felix, he quickly got back into a number of counts," Boston manager John Farrell said. "First pitch strikes were a little elusive, but he was able to go to a breaking ball or a changeup to get the timing back in his delivery. Just a workmanlike seven shutout innings."
It was Boston's seventh win in nine games, completing a 5-2 homestand. The AL East leaders play 16 of the next 19 away from Fenway Park, beginning Monday night in Houston.
Koji Uehara, the third Boston reliever, closed out the combined eight-hitter.
Diamondbacks starter Brandon McCarthy (2-5), activated before the game after being on the 15-day disabled list with right shoulder inflammation since May 31, gave up two runs on five hits in 4 1/3 innings.
Cody Ross and Martin Prado each had two singles for Arizona, which lost for the sixth time in eight games.
"They pitched good but, you can't give them too much credit," Diamondbacks manager Kirk Gibson said. "We had eight singles today and didn't put anything together."
Doubront, who experienced control troubles in a number of starts last year, had a simple reason why he's throwing more strikes.
"A lot of playing catch, a lot of front toss and a lot of that, that makes me be more consistent with pitches," he said.
Former teammate Ross saw a big difference.
"Yeah, a little bit more mature. He was good last year for a rookie," he said. "He pitched well. Today, we had him on the ropes and couldn't get anything when we needed to. He made some good pitches and located the ball well, but we had our opportunities and we blew it, and they didn't."
Boston grabbed a 2-0 lead and chased McCarthy with two runs in the fifth. Ellsbury's sacrifice fly made it 1-0 after consecutive singles. Shane Victorino then singled just over the head of McCarthy -- a scary image since the right-hander was hit in the head by a line drive last year -- before Pedroia's RBI double made it 2-0. David Ortiz was intentionally walked, loading the bases, but Will Harris relieved and got the next two hitters to pop out.
While with Oakland last September, McCarthy sustained what were described at the time as life-threatening injuries when he was struck in the head by a line drive off the bat of the Angels' Erick Aybar. The pitcher had an epidural hemorrhage, brain contusion and a skull fracture. Emergency surgery was performed that night and he was released from the hospital six days later.
Arizona right fielder Gerardo Parra had a tough day, dropping a fly ball in the first for and error and losing another in the sun for Ellsbury's RBI single that made it 3-0 in the sixth. Victorino followed Ellsbury's hit with a run-scoring single.
McCarthy held Boston hitless until Mike Carp's two-out single in the fourth.
Boston reliever Matt Thornton left the game in the eighth inning with tightness in his right oblique area.
"Thornton's dealing with a little bit of a right oblique issue." Farrell said.
The Diamondbacks had a singles in four of the initial five innings against Doubront, but thanks to two double plays and solid pitching none of the runners reached second.
Doubront has given up more than three runs just once in his 20 starts this season. ... Farrell said right-handed knuckleballer Steven Wright will make his first career start on Tuesday at Houston. "The reason being is that we're just trying to get guys an extra day of rest this time through," Farrell said. ... The Diamondbacks designated left-handed reliever Tony Sipp (3-1, 3.86 ERA in 48 games) for assignment before to game. ... Farrell turned 51 on Sunday. ... RF Victorino hit his left hip hard on the top of the wall, attempting to make a leaping grab of A. J. Pollock's foul ball in the third. He walked back to his position slowly, but stayed in the game. ... John Lackey (7-8, 3.23 ERA) pitches for Boston against Houston's Brett Oberholtzer (1-0, 3.77 ERA) in the opener of a three-game series on Monday. ... Wade Miley (8-8, 3.68 ERA) is slated to go for the Diamondbacks when they open a two-game set against the Rays and Jeremy Hellickson (10-4, 4.60 ERA) on Tuesday.