Borbon had four hits and four of Texas' eight stolen bases, Andrus swiped three more and the Rangers cut Boston's AL wild-card lead to a half-game with a 7-2 victory over the Red Sox.
"We were trouble last year," said Andrus, who has 23 stolen bases for Texas this season. "We're not afraid. That's what we like to do. Anytime we get on base, our goal is to get another base. That's our game."
The club-record eight stolen bases matched the most in the majors this season. Boston also allowed eight against Tampa Bay on May 3. The Rangers had six steals against the Red Sox on July 22.
Boston has only caught eight of 111 would-be base stealers this season. Catcher Jason Varitek took the blame for the Rangers' successful running game.
"I'm trying to get outs and make good throws," Varitek said. "I'm making the best throws I can make."
Borbon came into the game with three career hits and one stolen base. Rangers manager Ron Washington moved Ian Kinsler down to sixth in the order and inserted Borbon into the leadoff spot.
"It's an opportunity to show I can do some things at the top of the order and make things happen," Borbon said. "When you can combine speed and power, it's good. We get on base and our big boys can drive you in."
The Rangers bounced back after giving up six runs in the ninth inning of an 8-4 loss Friday night, blowing a chance to take the lead in the wild-card chase.
"So much for our balloon popping," Washington said. "That's been their makeup from day one. They deal with adversity, they don't let adversity handle them."
Two rookie pitchers also delivered solid performances for the Rangers. Derek Holland (6-7) allowed two runs in 6 2/3 innings to win for the third time in four starts. Neftali Feliz struck out three in two innings to pick up his first career save.
"We're not afraid of bringing our young kids up here and letting them play," Washington said. "The whole organization gets a pat on the back for that."
Kinsler, who also stole a base, got the Rangers off to a good start in his first plate appearance since coming off the disabled list Friday.
Leading off the second, Kinsler drove a 1-2 pitch from Penny barely over the fence in right. Andrus tripled two outs later and scored on Borbon's single.
After Bay hit his 25th homer on the first pitch of the fourth, the Red Sox went on to put two runners on with one out in the inning. Holland responded by striking out Brian Anderson and Alex Gonzalez to end the threat. The 22-year-old left-hander pumped his fist and yelled after escaping the jam.
The Rangers ran their way to two runs in the fourth. Andrus, who had three hits, singled and stole second and third on consecutive pitches before Borbon singled to make it 3-1. Borbon stole second and moved to third on Michael Young's grounder. Marlon Byrd then hit a sacrifice fly.
Andrus and Borbon each singled and stole a base in the eighth. They scored on Josh Hamilton's two-run single.
"That's part of their game," Boston manager Terry Francona said of the Rangers' speed. "They're not as effective when they don't have the lead."
Kinsler stayed in the game after he was hit in the helmet on a pitch from Fernando Cabrera in the eighth. He bounced up off the ground quickly and exchanged words with Varitek on his way to first base. ... Red Sox OF J.D. Drew didn't play because of a sore groin. ... Texas closer Frank Francisco got the night off after blowing the save Friday night. Francisco was pitching in a second straight game Friday. ... A day after he was acquired from Cincinnati, Gonzalez was 0-for-4 in his first start with Boston.