Brantley belted a 3-2 pitch from Phil Klein (0-1) into Texas' bullpen in right field and was mobbed by his teammates when he reached home plate after the Indians' seventh win of the season on a game-ending hit.
The Indians, who were 2-51 when trailing after eight innings, tied the game in the ninth on David Murphy's two-run homer with one out off of Neftali Feliz.
Scott Atchison (5-0) retired the final batter in the 12th.
The win gave the Indians (56-55) a three-game sweep and moved them over .500 for the first time since July 23 as they try to stay in playoff contention.
Texas has lost six straight on the road and has dropped 19 of 22 away from home. The Rangers (43-68) have the worst record in baseball.
Yu Darvish allowed one in seven innings, but saw an opportunity for his 11th win go down the drain. The All-Star right-hander held the Indians to four hits and struck out eight.
Feliz retired Lonnie Chisenhall to begin the ninth, but pinch-hitter Chris Dickerson walked. Murphy, who had a big series against his former team, hit the first pitch to right field.
Murphy had an RBI double in the second and was 4 for 4 with two RBIs Friday.
Neal Cotts pitched a perfect eighth before Feliz, who took over the closer's role when Texas traded Joakim Soria to Detroit, blew his first save in four chances.
The Rangers didn't produce much offense, but it appeared to be enough with Darvish on the mound. Alex Rios drove in a run with a first-inning single while sacrifice flies by Rougned Odor and Shin-Soo Choo added two in the second.
Trevor Bauer allowed three runs in a career-high 7 1-3 innings for Cleveland.
Elvis Andrus, continuing his torrid hitting against the Indians, blooped a double to left in the first and scored on Rios' hit.
Andrus extended his Progressive Field-record hitting streak to 25 games and has at least one hit in 43 of his 44 career contests against Cleveland.
Rangers: Left-hander Derek Holland, who has yet to pitch this season because of surgery on his left knee, will make his second minor league rehab start Monday at Triple-A Round Rock. He pitched two scoreless innings for Double-A Frisco on Wednesday.
Indians: Indians designated hitter Nick Swisher was removed from the game in the ninth inning due to a sore right wrist. Swisher, who is batting .212, had struck out in all three of his plate appearances before Dickerson hit for him.
Rangers: RHP Nick Martinez (1-7) will start the opener of a three-game series in Chicago against the White Sox on Monday night. He allowed six runs in 5 2-3 innings against the Yankees in his last start. RHP Hector Noesi (5-8) will start for Chicago.
Indians: RHP Corey Kluber (11-6), coming off his first career complete game shutout, will start the opener of a two-game interleague series against Cincinnati at Progressive Field on Monday night. RHP Alfredo Simon (12-6) will start for the Reds.