Swisher didn't make excuses for his 0 for 4 night, but the always-positive first baseman credited Cobb with making good pitches when he had to.
Bourn, Swisher and Kipnis went a combined 0 for 12.
"Sometimes you run into a buzz saw like that," Swisher said. "It's tough. Cobb was really locating his pitches — his changeup and his breaking ball. We had some clutch opportunities, but just couldn't come through."
The Indians had spent all season coming back from adversity. They reeled off 10 straight wins to end the regular season, and before that had 11 walk-off victories, with nine of them by different players.
They finally won over skeptical Cleveland fans, who turned out in force for the first playoff game in Cleveland since Oct. 18, 2007, when they lost Game 5 of the ALCS to Boston, then managed by Francona.
For almost nine innings, red-clad, towel-waving fans tried to coax one more comeback out of the Indians, who couldn't deliver again.
"I wish we could have given them a better game," Francona said. "The support was fantastic. We'll see, we have some work to do in the offseason, take maybe an hour or two to rest, and then we'll get back to work. It was pretty awesome to see how it can be, though."
Cleveland rookie Danny Salazar gave up three runs in four innings. He wasn't able to match Cobb, who missed a chunk of the regular season after he was hit in the head by a line drive.
Delmon Young homered in the third inning off Salazar as the Rays, playing in their third city over four days, advanced to face the AL East champion Red Sox in the best-of-five division series starting Friday
Desmond Jennings hit a two-run double in the fourth off Salazar, who gave the Indians a glimpse of the future.