Drew broke an 0-for-17 slump with a single in the fourth. His run-scoring hit off Texas reliever Akinori Otsuka (1-1) tied the game at 4.
He came in hitting .227 with two home runs and 14 RBI after signing a 5-year, $70 million contract as a free agent in the offseason.
"Getting a big hit in a key situation like that is always nice," Drew said. "I hope things are coming around. I'm trying to grind out good at-bats and see what happens."
"I know he's been struggling and I think he's been feeling it," Boston manager Terry Francona said of Drew. "But he stayed at it and had a couple of big hits."
Otsuka came into the game with a 1.02 ERA and had been Texas' most reliable reliever this season. But he couldn't hold the lead as Boston swept a series at Texas for the first time since Aug. 20-22, 1973.
"We had our pitching lined up pretty well," Texas manager Ron Washington said. "I felt confident out there with two innings left. But this is baseball, you can't predict it."
Jason Varitek also homered for Boston, which has the best record in baseball. The Red Sox hold an 11½-game lead over second-place Baltimore, 12 games over Toronto and 12½ over the New York Yankees.
Texas -- which has the worst record in the American League -- has lost five games in a row and 10 of its last 13.
Joel Pineiro (1-0) got out of a jam in the seventh to pick up the win. With two men on, he struck out Sammy Sosa and got some defensive help when center fielder Coco Crisp made a diving catch of Frank Catalanotto's liner.
"The key was Coco's play," Pineiro said. "He picked me up, picked us all up. It gave us the momentum. When I first looked at it, there was nobody close to that. But that was a big catch, the biggest key to the game."
Sosa went 0-for-5 with two strikeouts in his pursuit to become the fifth player to hit 600 home runs. He has 598.
Teixeira hit a 453-foot three-run homer and Ian Kinsler added an RBI single in a four-run sixth that put Texas up 4-3. Teixeira's homer was the fourth-longest hit at Rangers Ballpark at Arlington.
The Red Sox were without designated hitter David Ortiz, who was not in the starting lineup because of sore hamstrings.
Texas starter Kameron Loe, won hasn't won since April 21, gave up three runs and seven hits in six innings.
Tavarez had won his last two starts, but allowed four runs and six hits in 5 2/3 innings. Tavarez, who struck out six and walked one, yielded only Kenny Lofton's leadoff single in the first five innings before running into trouble in the sixth.
Drew singled to lead off the fourth. Lowell singled to left before Varitek lined a 1-1 pitch from Loe just over the fence in right for his fourth homer.
- Tavarez fielded a grounder by Catalanotto and ended up rolling the ball to Youkilis at first for the out. After the play, Youkilis was laughing behind his glove.
- Texas RHP Kevin Millwood gave up one hit in five scoreless innings on a rehab start for Double-A Frisco on Sunday.
- Gagne allowed his first run in 11 appearances this season.