Mujica served up the two-run homer to Gabe Gross that allowed the Rays to tie it, and Kobayashi gave up the three-run bomb by Carlos Pena that ended it.
"The team fights to put themselves in a position to win the ballgame," manager Eric Wedge said, "and you hate to see it ripped away that quickly. This was a tough day."
Tough and bizarre.
Hours earlier, Wedge yanked designated hitter Ryan Garko for standing in the batter's box and watching as his grounder down the first-base line stayed fair for an easy out.
Wedge also watched his second baseman, Asdrubal Cabrera, suffer a left ankle sprain while running to his position before the start of the bottom of the sixth inning.
But the Indians remained in a position to win because of Jhonny Peralta's second 5-for-5 game of the season, Jeremy Sowers' third consecutive quality start and two dominant innings of relief from Rafael Perez.
Sowers had the Tribe, bolstered early by Franklin Gutierrez's leadoff blast off Scott Kazmir, in a 3-1 hole after the first. He served up an RBI single to Evan Longoria and a two-run homer to Dioner Navarro.
But Sowers settled in after that, despite possessing what he called "lackluster command."
"I caught a couple breaks," Sowers said. "I left the ball up a couple times and got some flyouts. My command was not quite there."
The Indians, though, regained command of the ballgame with their bats.
Peralta blasted his 19th homer of the season with a solo shot in the third, then doubled to put two runners in scoring position with one out in the fifth. Shin-Soo Choo followed the double with a single to bring in one run, and Andy Marte, who had replaced the benched Garko, doubled to bring in two more.
Just like that, the Indians were back on top, 5-3.
And when the Rays threatened with a run off Sowers in the bottom of the fifth, Peralta responded right back with an RBI double in the sixth.
"I got a lot of good pitches to hit," said Peralta, who also had five hits on June 30 of this year. "I felt good."
The Tribe's good vibes continued when Perez came on in the seventh and tossed two scoreless innings to keep the Rays at bay. And when the Indians added another insurance run in the ninth, when Choo scored from third on a wild pitch, it appeared the Tribe was well on its way to a series win over the American League East's top club.
Then came Mujica.
In recent weeks, Mujica had been one of Wedge's most reliable options in an otherwise unreliable 'pen. But he had nothing but trouble in this brief yet memorable outing.
Mujica's misery began with Jason Bartlett's leadoff double on a liner to left. It continued when Eric Hinske ripped a grounder down the first-base line and turned it into an RBI double. And it reached its boiling point when Gross sent a two-run shot 437 feet over the right-field wall to tie it up at 7.
With that, Mujica was out of the game. And he was long gone from the locker room by the time reporters arrived.
Wedge, a manager without a closer, has had to get creative in the ninth inning, and his creativity came back to bite him in this one.
"We gave him an opportunity," Wedge said of Mujica, "and he wasn't able to get anybody out."
Neither was Kobayashi, who has quite a bit of ninth-inning experience under his belt. Akinori Iwamura legged out an infield single to first that Wedge protested to no avail. Kobayashi then walked Ben Zobrist before leaving a fastball up to Pena, who crushed it out to deep center.
Here at Tropicana Field, this game could be remembered as one of the highlights of a season in which the Rays may get their first whiff of October baseball.
For the Indians, it was just another sad reminder that their relief corps has been corrupted.
"Obviously," Wedge said, "we have a lot to work out in our bullpen."