Oakland, which began the night 12th in the AL in home runs, matched its season high for homers and extended its lead to 3½ games over Texas for second place in the AL West.
"This is a great place to hit in," said Swisher, who is 5-for-11 with two homers and eight RBI in the first three games of the four-game series. "There's no question about it. It's unbelievable here."
Swisher extended a streak that has coincided with his team's surge. Oakland has won 21 of 26, and the rookie has 11 homers and 31 RBI in his last 31 games.
"It's all clicking right now," Swisher said. "The pitching, the hitting, everything's working. It's really nice to see."
"Things aren't going our way right now," Teixeira said. "We need to win a lot of games. The teams around us aren't going to fold and let us take over. We know what we're up against."
Kirk Saarloos (6-6) allowed four runs and seven hits in 5 2/3 innings to help the A's beat the Rangers for the sixth time in seven meetings since the break.
"Those guys are helping us win games," A's manager Ken Macha said. "We have a lot of depth out there. When you come down here, you expect to use your bullpen a lot."
The A's jumped on Rodriguez early with a solo homer from Swisher in the first and consecutive shots from Johnson and Payton in the second for a 3-0 lead.
Texas came back in the third on Teixeira's three-run drive. He had been 6-for-36 since being the AL's starting first baseman in the All-Star Game.
The A's went back in front 4-3 on Chavez's 15th homer leading off the fourth and then got an RBI groundout from Bobby Crosby in the fifth to stretch their lead to 5-3.
Soriano's shot in the sixth pulled Texas to 5-4.
Rodriguez (2-2) gave up five runs and five hits in 5 2/3 innings. He struck out three and walked four.
"My pitches were high in the strike zone and when you do that they're going to kill the ball," Rodriguez said. "It's bad. We're losing. The pitchers aren't doing what they're supposed to do."
The Rangers fell to .500 (48-48) for the first time since they were 15-15 on May 6.
"We weren't good enough to match their pitching," manager Buck Showalter said. "Solo homers killed us. And the only reason we were even in that game was because they were solo homers."
James Baldwin, added to the roster before the game after being claimed off waivers by Texas from Baltimore on Thursday, entered in the sixth. Baldwin allowed one hit in 2 1/3 shutout innings.
Bobby Kielty's 14-game hitting streak ended. ... All six of Johnson's homers this season have come with the bases empty. ... Oakland got back-to-back homers for the first time this season. ... Texas' Hank Blalock was the DH for the first time in his major-league career. ... Soriano hit his 150th career homer.