Rookie winning pitcher Evan Scribner (2-0) left the mound in the sixth to a standing ovation from the sellout crowd of 30,067. He allowed two hits and struck out two in three scoreless innings after replacing struggling starter A.J. Griffin.
Ryan Cook, pitching for a fifth consecutive game, gave up a double to Nelson Cruz before retiring the next three Texas hitter, including strikeouts of David Murphy and Mike Napoli. Catcher Derek Norris pumped his right arm as the Coliseum fans jumped to their feet.
Norris then homered leading off the bottom of the eighth for his second RBI. It was his seventh homer and Oakland's majors-leading 112th since the All-Star break.
"Ever since Day 1 I've been here, it's been, the A's can't compete with the payroll, can't compete with this team or that team," Norris said. "We're better off if we're down. It just gives us the extra energy."
The A's join the NL West champion San Francisco Giants as division champions. The Bay Area is already buzzing about a possible Bay Bridge World Series like the earthquake-interrupted 1989 championship swept by Oakland.
Hamilton's miscue while charging forward might haunt the to-be free agent if his Rangers don't get past their wild-card game.
Murphy's two-run single highlighted a five-run third inning that put Texas in prime position.
"You can have all the experience as you want but when you run into a team that's hot, experience has nothing to do with it," Texas manager Ron Washington said.
The only other teams to come back from at least 13 games down to win the division were the 1914 Boston Braves, the 1951 New York Giants, the '78 Yankees and the '95 Seattle Mariners.
Oakland accomplished this with an ever-changing roster managed by Melvin in his first full season as skipper. They lost third baseman Scott Sizemore to a knee injury on the first full day of spring training workouts, never promoted slugger Manny Ramirez from the minors before parting ways, and dealt with devastating injuries all year long.
Opening day starter Brandon McCarthy took a line drive to the head Sept. 5 that required surgery and ended his season, Brett Anderson missed most of the year coming off Tommy John surgery, and Dallas Braden never pitched because of shoulder problems. Starter Bartolo Colon received a 50-game suspension in August for a positive testosterone test.
Third baseman Brandon Inge needed shoulder surgery last month and prized Cuban rookie Yoenis Cespedes missed time with a pair of injuries in May and June.
And that's just the beginning for a team that traded away catcher Kurt Suzuki to the Nationals during the year after swapping three top pitchers during the offseason — Trevor Cahill to Arizona, NL Cy Young Award favorite and 21-game winner Gio Gonzalez to Washington and All-Star closer Andrew Bailey to Boston.
"It's kind of like the story of our whole season. Nothing has been easy," reliever Sean Doolittle said. "End of June, we were 13 back of those guys. We've had injuries, we've had a lot of young guys step up, we've had a lot of guys coming up and down. We were never out of it. You could feel the vibe."