Landon Powell drew a walk off Dustin Nippert (1-3) to lead off the 13th and stole second base. After Rajai Davis struck out, Powell advanced to third on Cliff Pennington's ground out and came home when Barton shot a grounder past second baseman Ian Kinsler.
Tyson Ross (1-0) earned his first major league win by hurling a scoreless bottom of the 13th.
Barton also homered in the 11th to give the A's a brief lead before Ross gave up the tying run in the bottom of the frame on Julio Borbon's RBI single.
Eric Chavez socked a two-run fifth-inning homer in the A's win.
The Rangers got a two-run homer from Josh Hamilton. David Murphy and Borbon each collected a pair of hits and drove in a run in the Texas loss.
Boston schools pull out free condoms over wrapping complaints
Millions of Getty images now available for free via embed tool