The Rangers' eighth win in their last nine games reduced their magic number to three for clinching the division. The Rangers remained five games ahead of the Angels, who won at Toronto Wednesday night.
On a day the Athletics stripped the interim tag from manager Bob Melvin and rewarded him with a three-year contract through the 2014 season, the host team failed to hold onto a late lead and lost for the third straight time.
Kinsler led off the eighth with his 30th homer of the year, a shot to left field on a full-count offering. One out later, Hamilton went deep to right field off Balfour (4-2).
Mike Adams threw a perfect eighth inning after Koji Uehara (2-3) held the A's down in the seventh. Neftali Feliz walked Josh Willingham leading off the bottom of the ninth, but retired the next three hitters to lock down his 29th save.
"It was fun, a little stressful, but fun," Hamilton said.
Heading to the hill for Texas this afternoon will be righty Colby Lewis, who has already beaten the A's four times this year and will try to become the first Rangers pitcher to beat them five times in a season since Fergie Jenkins in 1974.
Lewis has won his last two starts overall and is 13-10 on the year with a 4.45 ERA.
Oakland, meanwhile, will counter with righty Trevor Cahill, who is 11-14 with a 4.31 ERA. Cahill had a two-start winning streak stopped versus Detroit on Friday, as he allowed three runs and nine hits in 6 2/3 innings.
Cahill beat the Rangers the last time he faced them and is 9-4 lifetime against them with a 2.69 ERA in 14 starts.
Texas is 13-5 versus the A's this season.