Texas won the first game 11-9 and the nightcap 10-5 to pull within two games of Boston for the AL wild card. It's the closest the Rangers have been to a playoff spot this late in a season since they won the AL West in 1999. Boston beat Baltimore 10-0.
Byrd raised his average 10 points in the two games to .286. He went 4-for-4 in the first game, including the home run, and 3-for-5 in the nightcap.
Brandon McCarthy (7-2) started the second game and didn't allow a hit until the fourth, but tired quickly. He was making his second start after spending three months on the disabled list with a stress fracture in his right shoulder blade. McCarthy was charged with five runs in 6 1/3 innings, allowing a solo homer to Kelly Shoppach in the fifth and a two-run blast to Asdrubal Cabrera in the sixth.
Indians starter Aaron Laffey (7-5) lasted just 3 1/3 innings, equaling the shortest start of his career. He was charged with seven runs and 12 hits.
Texas' Chris Davis went 3-for-4 in the nightcap, including a three-run homer.
In addition to Borbon and Byrd, David Murphy homered in the first game for Texas, while Matt LaPorta and Travis Hafner homered for Cleveland.
Borbon now has three homers in 70 major league at-bats after hitting just nine in 1,091 career minor league at-bats.
"He's capable of doing that," Rangers manager Ron Washington said. "I don't know if you'll see him hit 15 or 20 this year, but he certainly has a line drive stroke and sometimes he can catch it and hit it out."
Indians pitcher Chris Perez and Texas' Neftali Feliz entered the day first and second among relievers with consecutive scoreless innings streaks. Both of them ended in the first game, although Feliz (1-0) picked up his first major league victory in the process.
Perez had gone 20 2/3 innings, the best in baseball, before Byrd pounded a 2-1 pitch into the bleachers in left in the seventh, breaking a 5-5 tie. Texas led the rest of the way.
It was the first run Perez allowed since July 7, a span of 20 appearances.
"He's done a tremendous job for us, but not today," Indians manager Eric Wedge said. "You're not going to be perfect, that's for sure."
Feliz's streak was next to go thanks to Hafner's solo homer in the seventh, ending his at 20 1/3 innings.
"He's only human," Washington said of Feliz, who had retired 18 consecutive batters. "I don't think I can remember any pitcher that came to the big leagues and had a 0.00 ERA. He's going to give up something."
Jensen Lewis (2-4) pitched 1 1/3 innings to take the loss.
An estimated crowd of less than 1,000 witnessed the start of the first game. Monday's rainout pushed the start time up two hours, impacting the late-arriving crowd. The official attendance for the day was 12,976.
Notes
Rangers shortstop Elvis Andrus extended his hitting streak to 11 games. … Designated hitter Andruw Jones, activated off the DL earlier in the day, went 1-for-4 in the nightcap. … Indians RHP Joe Smith, who hasn't pitched since Aug. 29, is battling a sore left knee. He is hoping to return sometime next week. … Center-fielder Michael Brantley is the second Indian in the last 50 years to hit safely in each of his first eight games. Josh Bard hit in his first nine in 2002. … Catcher Lou Marson made his major league debut in the first game, going 1-for-4. … The Indians purchased the contract of infielder Niuman Romero from Triple-A Columbus. … The Indians and Class A Lake County Captains have agreed on a four-year extension to their Player Development Contract (PDC) through the 2014 season.