Shin-Soo Choo was particularly good off Cobb during his two at-bats that resulted in a pair of doubles, two runs and an RBI.
Cobb has now lost six of his last eight starts after winning his first two.
"My changeup I was able to throw a little bit for a strike, but my fastball, curveball were just all over the place," Cobb said. "If you don't locate your fastball and get ahead, you have no chance really to last late in the game."
Tampa Bay used five relievers, who kept the Rays in the game by not allowing a run and giving up just three hits and three walks to go with seven strikeouts.
"Alex did struggle, that was the bad part of the whole thing," Rays manager Joe Maddon said. "To have to use that many guys to piece that game together."
Maddon added after the game that relievers J.P. Howell and Kyle Farnsworth may not be good to go in Tuesday's game after pitching Monday, but the bullpen should still be fine because Burke Badenhop and Wade Davis, both of whom pitched Sunday, would be available again.
The defense also held strong throughout the night, tying a club record with five double plays, with Will Rhymes being involved with each.
The Rays' woes at the plate continued, as well. Though they drew six walks, the Rays were still only able to score twice and get six hits. Tampa Bay has scored three or fewer runs in six of its last nine games.
Choo led off the game for the Indians with a shot to deep center field that was originally ruled a home run. After the umpires checked the replay, it was determined that the ball hit the wall on padding just on the yellow line, forcing Choo to settle for a ground-rule double.
"After they made the call, we came in and watched the replay," Indians manager Manny Acta said. "It hit the yellow line and came back. At the beginning, from the dugout, we thought that it might've hit the top and then the back where it has that sign. But it was the right call."
After a pair of groundouts by Cleveland hitters, Choo found himself on third base, and a ball that got past catcher Jose Molina allowed him to score the game's first run.
"It happens early in the game, so you don't sometimes give it as much weight, but they all count," Maddon said about the play. "That was big. It looked like a curveball pretty much right there, and [Molina] just missed it. That's something he doesn't normally do, and I know he's upset about it."
Choo came through again for the Indians in the third when his second double of the game scored Johnny Damon, who doubled to lead off the inning. Two batters later, Jason Kipnis doubled home Choo to make it 3-0.
Tampa Bay got on the board in the fourth with a quick start. B.J. Upton hit a leadoff single, and Luke Scott singled him over to third. Jeff Keppinger then lined a hit off Indians starter Zach McAllister's right forearm that went for a single and scored Upton.
Desmond Jennings, batting near the bottom of the order for the fifth straight game, hit a homer over the left-field fence to pull the Rays to within one to lead off the seventh.
"I'm definitely seeing him mentally working better at-bats," Maddon said. "That's the most important thing. He's getting close to being back [at leadoff]. We'll see."
The Rays nearly scored again in the eighth, with Keppinger batting and Scott on second with one out. Keppinger hit a blooper to right field that was barely foul, which likely would have scored Scott if it were fair. Instead, Keppinger flew out to right field, and after Jennings' single sent Scott to third, pinch-hitter Brooks Conrad struck out to end the inning.
"Keppinger's ball was one centimeter from tying up the game and possibly to the point where we could go up," Scott said. "One little millimeter or centimeter, the game's changed."