Without Ramirez, the Red Sox managed to score 15 runs in wins on Saturday and Sunday.
"If we have a good approach, we'll live with it either way," Red Sox manager Terry Francona said. "When balls leave the ballpark, I think [they] sometimes are by-products of having good at-bats, swing at strikes, get good, aggressive swings at strikes and take balls. And some days the ball jumps out of the ballpark, but if we have that approach, we're going to be OK either way."
Beckett (7-4) scattered six hits while fanning six. Beckett, who homered in Philadelphia in 2006, went hitless in three at-bats but did lay down a sacrifice bunt.
Beckett said he was never concerned about the Sox making up for lost offense without Ramirez.
"I don't worry too much about that," Beckett said. "I know these guys are going to take care of their part. I just want to go out there and do what I'm supposed to do, go deep into games. I felt like we did a pretty good job today."
Boston blasted Cincinnati starter Homer Bailey (0-3), tagging the right-hander for five runs and three homers over his 2 1/3 innings.
The Red Sox jumped on top in the first, when Ellsbury singled, stole second and third for his 32nd and 33rd steals of the season, the first of which established a new franchise rookie single-season mark. He then scored on Pedroia's sacrifice fly.
"It's nice to get that momentum and get that run on the board to give our starter Josh a run to work with," Ellsbury said.
That would be all Beckett, David Aardsma and Mike Timlin would need on this day. But that didn't keep the Red Sox from adding on.
Crisp connected for his fourth homer and second in as many days with Jason Varitek aboard in the second to make it 3-0.
"Who cares about Beckett? Let's talk about me," Crisp joked. "Nah, Beckett did a great job. The game that he pitched today is the typical Beckett we were able to see last year, and if he's able to keep building off today, he'll be unstoppable again."
Ellsbury, on a curveball, and Drew, on a middle-in fastball, connected for solo homers off Bailey in the third, and the rout was on. It was Ellsbury's fifth and Drew's 11th.
Meanwhile, Beckett's lone nemesis on this day was Joey Votto. The Reds' rookie first baseman had three of Cincinnati's six hits on the day, all off Beckett.
Beckett stranded two runners in the first two innings and his last inning, the seventh, but made big pitches each time to preserve Boston's sixth shutout win this year. His final pitch generated an inning-ending 4-6-3 double play off the bat of Jay Bruce in the seventh.
"I thought Josh set the tone," Francona said. "He threw 22 out of 28 first-pitch strikes. He attacked the strike zone. [The] first couple of innings, he had to pitch with some runners on base, but he made pitches. And then we swung the bats early real well and then spread it out and had a good day at the ballpark."
"You're always trying to get ahead," added Beckett. "That first pitch is key. Two of the first three [pitches], and then you work off that -- 0-0 and 0-1 counts are big."
When Pedroia connected for his fifth homer, a liner to the bleachers in left in the sixth, it gave the Red Sox home runs from the top three in their order for the first time since July 10, 1997, when Nomar Garciaparra, John Valentin and Mo Vaughn all went deep.
"That's pretty cool," said Ellsbury. "When the top of the lineup is swinging like that, we're going to win some games, not just with our legs, but our bats as well."