"We got the win," manager Ron Gardenhire said. "But we took a beating doing it."
Lew Ford's hustle turned his base hit off Rafael Betancourt (1-2) into a one-out double in the 13th. Two batters later, Jones hit a liner to right and Ford raced around third, beating Casey Blake's throw to the plate.
Morneau tweaked his left elbow while hitting balls off a tee before the game and was replaced after going 0-for-2 against Indians starter Scott Elarton. Morneau was taken for X-rays, which came back normal and he is listed as day to day.
Punto is headed to the disabled list after pulling his right hamstring while trying to steal second base.
"It'll be awhile for him," Gardenhire said.
Mauer aggravated a groin strain that kept him out last month. He was removed from the lineup in the 12th inning.
Six of the game's seven runs were scored in the first five innings. After that, pitching took over, with the Twins managing just four hits over the final eight innings and the Indians two over the final nine.
Cleveland managed only four hits against Santana, last season's AL Cy Young award winner, but scored three runs thanks to homers by Victor Martinez and Coco Crisp, who was activated from the disabled list before the game to replace an injured Juan Gonzalez, and a triple by Ben Broussard.
"He hung two sliders," pitching coach Rick Anderson said. "He threw the ball very well and did a good job of recognizing what was working and what wasn't."
Other than those big hits, however, the Indians did nothing against Santana, who ran his major league-leading strikeout total to 105. He also struck out 14 on Sept. 19 last year against Baltimore.
J.C. Romero (1-2) earned the win with an inning of scoreless relief, helping the Twins take two of three from their AL Central rivals.
Crisp went 1-for-4 with a solo homer and a sacrifice bunt for the Indians.
Gonzalez, a two-time AL MVP who has missed most of the past three seasons with injuries, strained his right hamstring in the first inning of Tuesday's game while trying to beat out a slow grounder in his first major league at-bat in more than a year.
He was placed on the 15-day disabled list Thursday.
Team officials said they were unsure how much time he would miss. He missed nearly three months with his last hamstring injury.
While not as overpowering as Santana, Elarton was just as sharp. He allowed just one run and three hits through the first four innings before Punto's two-run homer in the fifth tied the score at 3.
Elarton scattered five hits over seven innings, allowing just three runs and striking out three.
"I was in a position to do better than I did," Elarton said. "That's the disappointing thing."
The Indians jumped on Santana in the first inning when Martinez hooked a curveball just inside the foul pole in left field for a 1-0 lead. Broussard tripled past a diving Shannon Stewart in left on the next pitch and scored when Ronnie Belliard lined a single just past Santana's head.
Santana struck out his next four hitters and retired nine in a row. He needed just 11 pitches to strike out the side in the second and yielded just one hit the rest of the way, a homer to Crisp in the fourth that gave Cleveland a 3-1 lead.
Crisp had a chance to rob Mauer of a solo homer in the first, but he hit the wall unexpectedly, and the ball glanced off his glove.
"It was right there," Crisp said. "When I reached up, the baggie hit my shoulder blade and stopped my arm. It was an easy play. I didn't even have to jump."
It was yet another close ballgame between the teams. In 13 meetings this season, 11 have been decided by three runs or fewer, including eight one-run games and four that went extra innings.
"Us and the Twins, we've had tough games all year long," Cleveland manager Eric Wedge said. "It seems like every game has been tight."
- The Twins recalled Terry Tiffee from the minors to replace Punto.
- Minnesota's 18 strikeouts set a club record for an extra-inning game.
- Broussard had homered in consecutive games. He went 1-for-5.
- Leadoff hitter Jody Gerut was the only Indians player that did not strike out against Santana. He was 0-for-5 with a walk and struck out against Romero in the 13th.