Nelson Cruz -- who finished a single shy of the cycle -- and David Murphy had three hits apiece for the Rangers (33-23), who finished 3-3 on their six-game road trip through New York and Boston.
"We'll take what we can get," manager Ron Washington said. "We had been coming in here [to Boston] and not having much luck, but we came up here and played two solid ballgames and got away with the [series] win.
"I'm very happy with the way our guys recovered [from Saturday's 8-1 defeat]. After every loss we've had, they've always recovered. That just shows how much character we have."
C.J. Wilson nailed down his fifth save of the season in the absence of Frank Francisco, who was unavailable for the third straight game after complaining of right shoulder soreness before Friday's series opener. Washington said the closer will be re-evaluated on Monday, when the Rangers return home to open a four-game set against Toronto.
One night after spoiling Jon Lester's perfect-game bid, Michael Young put Texas on top in the first with a solo home run to right field. The third baseman pummeled a high fastball from starter Daisuke Matsuzaka (1-4) into the Boston bullpen.
Omar Vizquel singled home Murphy in the second, lining a ground ball through the left side of the infield. Murphy, who once played for the Red Sox, extended the Rangers' cushion to 4-0 in the following frame with a two-run double off the center-field wall.
Jacoby Ellsbury halved the Boston deficit in the bottom half of the inning, scalding a ground ball to second base that eluded the backhand of Ian Kinsler and kicked into center field. Jason Varitek and Mark Kotsay scored on the play, and Kinsler was charged with an error.
Kotsay trimmed the Rangers' lead to one run in the fourth with a solo shot to right, but Boston (33-24) struggled to generate offense the rest of the way against Padilla (4-3) and relievers Darren O'Day and Wilson.
Cruz played an instrumental role in Texas' final two runs, scoring from third on a Matsuzaka wild pitch in the fifth before tattooing a fastball from Hideki Okajima over the Green Monster in the seventh.
The efforts of Cruz and the Rangers' offense helped lessen the burden on Padilla, who said he did not feel any added pressure to perform in the wake of this past week's events.
"If you start thinking about that, then you're going to get nervous and not do your job," Padilla said. "I just have to keep doing what I have to do, and that's it.
"The guys gave me run support, so I got more comfortable on the mound."
Washington was impressed with how well the veteran righty handled a potent Red Sox lineup.
"It all centered around Padilla going out there and pitching well," Washington said. "He did a great job keeping those guys off balance throughout the game. Not many people can do that to those guys. They usually find a way to make pitchers work and get them out of the game.
"But today, he gave us exactly what we needed."