Hunter's double into the left-centre gap came off Joe Nathan (3-5), who blew a save for only the third time in 39 chances this season.
The big hit by Hunter came right after Nathan struck out Mike Trout, who earlier in the game became the first major league rookie with 30 homers and 40 stolen bases in the same season.
Los Angeles (88-70) closed two games behind Oakland (90-68) for the AL's second wild card. The A's played at home later Sunday against Seattle.
Texas (92-66) saw its division lead over the A's cut to two games with four to play.
With the loss, the Rangers won't have an opportunity at home to clinch their third consecutive AL West title. They close the regular season in a three-game series at Oakland that starts Monday. The Rangers would assure a playoff berth with a victory against the Angels on Sunday night.
Los Angeles trailed 4-3 in the ninth before a one-out single by Maicer Izturis, retired on a reversed call two innings earlier. Nathan walked Chris Iannetta, Trout struck out and Hunter, who lives in nearby Prosper, hit a drive that rolled to the wall.
Garrett Richards (4-3) retired Nelson Cruz with two runners on in the eighth, and Ernesto Frieri pitched a perfect ninth for his 24th save in 26 chances.
That wasn't what the Orioles and their fans wanted to see while watching the Rangers game on the scoreboard in centre field in Baltimore. An Angels' loss would have clinched the Orioles' first post-season berth since 1997.
Left-hander Derek Holland (11-6) was set to pitch the nightcap for Texas against right-hander Ervin Santana (9-12). That was the scheduled matchup for Saturday's game, which was called off four hours after its scheduled start because of rain.
Yu Darvish struck out seven in 6 2-3 innings but came out after Trout's homer, which trimmed the lead to 4-3, and Hunter followed with a double.
Alexi Ogando relieved and induced an inning-ending grounder on a full-count pitch to Albert Pujols, who already had two singles. With primary setup man Mike Adams injured, Ogando stayed on for the eighth.
Cruz hit his 24th homer for Texas, a two-run shot in the second, an inning after the right fielder threw out Trout on the bases.
Trout started the game with a walk and quickly got his 48th stolen base. His 30th homer, a solo shot to straightaway centre, came on the first pitch Darvish threw him in the seventh.
After the stolen base in the first, Trout tried to advance on Hunter's flyball to right. But Cruz threw a one-hop strike to third baseman Adrian Beltre to retiret Trout, who slid head-first into the same side of the base where Beltre was.
Izturis led off the seventh with a weakly hit ball that dropped between the mound and first base. Mitch Moreland fielded the ball, and tossed to Darvish, who made the catch, tagged the base and showed the ball to umpire Mark Wegner, who signalled safe.
Manager Ron Washington came out of the dugout to argue, but umpires huddled and called Izturis out, bringing out Angels manager Mike Scioscia. Two batters later, Trout homered.
Trout had the first of three consecutive two-out singles in the third and scored on Pujols' hit.
Elvis Andrus drew a two-out walk in the third and scored on Josh Hamilton's single that snapped the slugger's 0-for-17 career mark against Zack Greinke. Beltre hit a popup to shallow right field that Hunter never saw, and it dropped for an RBI double and a 4-1 lead.
NOTES: The 21-year-old Trout is the youngest player with a 30-30 season. He needs two stolen bases to become only the third player ever with 30 homers and 50 stolen bases. The others? Eric Davis (1987) and Barry Bonds (1990). ... Beltre made an incredible defensive play starting the eighth, diving to his right for a back-handed stab. He got up, barely pivoting his feet while making a quick sidearm throw that got Kendry Morales out at first. ... Hamilton, who struck out three times, has 126 RBIs. ... There were no weather issues Sunday for the first-ever split doubleheader at Rangers Ballpark.