"That was probably his best outing as far as his ability to command the strike zone," Boston catcher Jason Varitek said. "It's good. It seemed like our bats were a little slow. We needed a good pitching performance."
Pedroia extended his career-high hitting streak to 15 games with a double high off the Green Monster on the first pitch from reliever Brian Bass (3-3) in the eighth. After J.D. Drew advanced Pedroia to third, Ramirez lined a single to right through a drawn-in infield.
The Red Sox, coming off a 3-7 road trip that dropped them into second place behind Tampa Bay, won for the 23rd time in 29 games in Fenway.
Matsuzaka pitched 7 1/3 innings, allowing six hits, walking three and striking out five. Hideki Okajima (2-2) escaped a bases-loaded, one-out jam in the eighth for the win.
It was Boston's major league high 10th shutout.
For the first time in a few starts, Matsuzaka was pleased with his control.
"Of all our starting pitchers, I'm the only one who hasn't been able to get really deep into the games," he said through a translator. "So what I've been trying to do is go deep into the game and help out the bullpen."
The Twins lost just their fourth in 22 games, ending a five-game winning streak.
"A heck of a baseball game," Minnesota manager Ron Gardenhire said. "I think both teams had lots of opportunities all the way through. They finally came through and got a hit."
Jonathan Papelbon got the final three outs for his 26th save.
Baker gave up five hits, walked two and struck out seven in seven innings. He held an opponent to three runs or fewer for the 19th time in 20 starts, dating back to last season.
"Obviously, with them playing at home, they play very well here," Baker said. "I gave it everything I had. We made some great defensive plays, allowing me to continue to pitch and pitch into the seventh inning."
Matsuzaka, who hadn't pitched beyond five innings in his three starts since returning from the DL on June 21, worked out a bases-loaded jam he caused with his own wildness in the first. After loading the bases with two outs on a pair of full-count walks and an infield hit by Morneau, he got Young on a bouncer to the mound.
"His fastball was just electric and coming out of his hand real good," Twins right fielder Denard Span said. "He was hiding the ball good."
It was the fifth time in his brief career Matsuzaka has gone seven or more shutout innings. He has yet to record a complete game shutout.
Boston's best early bid came in the third, when Drew and Ramirez singled to open the inning before Mike Lowell fanned. After a walk by Kevin Youkilis, Coco Crisp and slump-mired Varitek were each retired easily.
Good defensive plays contributed to the low scoring game. Span made a running, semi-jumping catch of Crisp's deep drive in the seventh, and Boston's Jacoby Ellsbury made a diving stab of Carlos Gomez's eighth-inning liner.
Minnesota's Alexi Casilla extended his hitting streak to a career-best 14 with an eighth-inning bunt single.
Varitek, selected by the players to the AL All-Star squad, went 0-for-3 and is 6 for his last 63.
Red Sox DH David Ortiz, on the DL since May 31 because of a partial tear of the sheath that surrounds the tendon in his left wrist, took his first live BP before the game and said he felt "good." ... Boston optioned RHP Justin Masterson to Triple-A Pawtucket earlier in the day and recalled 1B Jeff Bailey. The club plans to convert Masterson into a reliever. ... Minnesota, coming off a three-game home sweep of the sagging Indians, opened a seven-game road trip Monday. ... The Twins won three of four games against the Red Sox in the Metrodome in May.