Felix Doubront allowed one run over six innings as the Sox beat the Indians, 4-1, before 38,048 at Fenway Park that enjoyed a crisp, drama-free night of baseball.

“It’s definitely great to see Felix come out and just dominate,’’ said Cody Ross, whose long home run in the sixth inning helped secure the win. “He was pounding the zone in and out with his fastball. Great command [and] his slider and curveball were great.’’

Doubront continued to be the most consistently effective starter in the rotation, allowing three hits and walking two in his first career appearance against the Indians. He struck out five.

The Sox would have preferred another inning out of the 24-year-old lefthander, but three relievers each handled one inning without any difficulties. It was the seventh save in nine chances for Alfredo Aceves.

Sox pitchers retired the final 11 Cleveland batters in a game that lasted a tidy 2 hours 30 minutes.

The Sox have won two of the first three games against the Indians and can take the series on Sunday afternoon with Daniel Bard on the mound against Justin Masterson.

“Two straight,’’ catcher Jarrod Saltalamacchia said. “Have to start somewhere.’’

Doubront (3-1) has a 4.46 earned run average and the Red Sox have won five of the seven games he has started. With each start, his confidence grows.

“Every start, they challenge me and I’m trying to do my best and learn from each start,’’ he said. “I’m learning and I’m getting better. I just need to throw more and that’s it. I’m just waiting for that moment to go more deep in the game.’’

Doubront was working on a one-hit shutout through five innings and had put only three runners on base.

The Indians got their second hit to lead off the sixth inning when Lou Marson hit a shallow pop to right field that bounced past a diving Ross for a double.

Marson moved to third on a ground out and scored on an infield single by Jason Kipnis.

Doubront ended the inning by getting Asdrubal Cabrera on a pop to second base after falling behind in the count and Carlos Santana on a grounder back to the mound.

Doubront did not let the situation overwhelm him.

“I wasn’t going to overthrow,’’ he said. “Just throw the ball right there, the hitter has to do something.’’

That was the end of the night for Doubront, who had thrown 109 pitches.

Lefty Andrew Miller retired all three batters he faced in the seventh inning. He has been near perfect in five innings since being activated from the disabled list May 6, putting two runners on base and not allowing a run.

Miller has simplified his mechanics, working only from the stretch and staying consistent with the angle of his arm. He has thrown 57 pitches since returning, 41 of them strikes.

“You still wonder when you go to the bullpen, but Andrew Miller has been fabulous,’’ manager Bobby Valentine said. “He continued his streak of throwing strikes with quality pitches.’’

Vicente Padilla took care of the eighth inning.

With Josh Tomlin going on the disabled list with a sore wrist, the Indians summoned 24-year-old righthander Zach McAllister from Triple A Columbus to start the game.

He struck out eight without a walk in seven innings but allowed all four runs on eight hits.

Mike Aviles, batting ninth, led off the third inning with a single to center. With one out, Dustin Pedroia hit a high fly ball to left-center.

As left fielder Shelley Duncan and center fielder Michael Brantley waited for the ball to come down, Aviles turned a few steps past second base and stopped.

When the ball scraped the wall, Aviles took off, rounded third, and beat the throw to the plate. Pedroia finished 2 for 4 and extended his hit streak to 13 games.

David Ortiz was next and his double to center field scored Pedroia. It was the 14th double of the season for Ortiz, who has 24 RBIs.

The lead grew to 3-0 in the fourth inning. Daniel Nava reached base for the eighth time in 11 plate appearances since being recalled when he singled to left field.

Jarrod Saltalamacchia pulled a changeup to right field. The ball hopped into the stands for a double and Nava had to stop at third base.

Aviles followed with a fly ball to left field. Nava tagged and beat a terrific throw from Duncan when he slid under the tag by Santana.

“I’m just glad to be a part of two wins,’’ said Nava, who was 2 for 3. “We had a great performance by Felix out there tonight and we were able to get a few runs across to make a difference for him. It’s fun to be a part of this.’’

Ross gave the Sox a 4-1 lead in the sixth with his first home run since April 23, a shot that cleared everything in left field and landed on the roof of the parking garage on Lansdowne Street.

It was his sixth of the season.

Peter Abraham can be reached at pabraham@globe.com. Follow him on Twitter @PeteAbe.