The Angels won, so the Twins couldn't clinch tonight.
The Twins twice jumped out to leads on a night in which starter Bartolo Colon left after one inning because of an illness.
Taylor Rogers (7-3), the eighth Minnesota pitcher, got the last two outs of the seventh inning for the win.
Matt Belisle worked the ninth for his eighth save in 10 tries since the Twins traded all-star closer Brandon Kintzler.
Trailing 6-4, the Twins rallied in the eighth inning against Shaw. First pitch was at 7:10 p.m. with a temperature of 82 degrees.
Minnesota added an insurance run on an RBI single by Byron Buxton in the ninth.
Meanwhile, Indians pitchers could not solve Rosario, who entered the game hitting.392 against the Indians with two home runs and five RBI. Robbie Grossman singled, with Goodrum advancing to third. Francisco Lindor's sacrifice fly drove in Kipnis, giving Cleveland a 5-4 lead. It was his 33rd home run of the season.
Each team scored two runs in the first inning. They were 40-2 when leading after seven innings. Dozier went to third on a groundout by Joe Mauer, and scored on a groundout by Jorge Polanco.
The Twins grabbed an early 4-2 lead off Indians starter Josh Tomlin thanks to Eddie Rosario's solo home run in the first and Rosario's RBI double in the third. Colon struck out Austin Jackson, but Jose Ramirez doubled, sending Lindor to third.
Minnesota took a 4-2 lead in the third. Mauer advanced to second when Polanco grounded out to Tomlin.
When Colon exited after one inning, the parade of Minnesota relievers began. Encarnacion, who had moved to second on a wild pitch, was thrown out at the plate on Bruce's single. Before he connected off Shaw, the Twins were 1 for 14 with runners in scoring position. Encarnacion also tried to score on the hit, but the throw home from right fielder Max Kepler arrived in plenty of time to retire Encarnacion, keeping the score 4-4. The 44-year-old right-hander gave up hits to three of the first four batters he faced and saw his season earned run average climb to 6.72. Chisenhall hasn't appeared in a game since September 14 because of a strained right calf, the second time this season he has been sidelined by that injury. Dozier scored his 100th run tonight (and his 101st and 102nd), putting him at the century mark for the fourth straight season. He becomes the fifth player in Indians history with 100 RBIs and 100 walks in the same season.
Encarnacion joins an exclusive list of Tribe sluggers that includes Travis Hafner, Jim Thome, Andre Thornton and Al Rosen to accomplish the 100/100 feat. It was his 23rd consecutive successful steal, breaking the Twins franchise record. Polanco has 72, trailing only Roy Smalley, who had 94 in 1979 and 77 in 1978.