Cobb (4-3) gave up three runs on six hits with seven strikeouts and one walk.
Cleveland starter Josh Tomlin (2-5) lasted just 2 2/3 innings, giving up six runs on seven hits with five strikeouts and no walks.
Tampa Bay hit 10 home runs in the three-game series, eight in the last two games, both of them wins. Wednesday's sluggers were Corey Dickerson, Logan Morrison and Jesus Sucre.
Tomlin had an odd outing.
After giving up a single to Dickerson leading off the game, Tomlin struck out the next five batters in a row. Then he gave up hits to the next four batters, retired two in a row, gave up hits to the next two hitters, and was removed from the game.
With two outs and nobody on base in the top of the second, Tim Beckham singled and went to third on a single by Daniel Robertson. Sucre's double scored Beckham, and Robertson scored on Dickerson's third home run in two days, giving Tampa Bay a 4-0 lead.
Morrison's solo home run off Tomlin in the third inning made it 5-0, and an RBI single by Beckham off reliever Dan Otero later in the inning extended the Rays lead to 6-0.
Cobb retired the first six batters he faced in order, but Cleveland scored twice in the third inning.
Lonnie Chisenhall led off with a double, and he scored on a double off the left field wall by rookie Bradley Zimmer. It was the the first major league hit and RBI for Zimmer, who was called up from Triple-A Columbus on Tuesday.
An RBI double by Michael Brantley in the fifth inning cut the Tampa Bay lead to 6-3, and Zimmer's first major league home run, in the ninth inning off Alex Colome, accounted for Cleveland's final run.
The game ended when center fielder Kevin Kiermaier robbed Jason Kipnis of a potential home run by making the catch just in front the fence.
NOTES: Rays 2B Brad Miller missed his second consecutive game due to a strained abdominal muscle. ... In Tuesday's game, 3B Evan Longoria broke Carl Crawford's franchise record for most career at bats in Rays' history. Longoria has 5,001 career at bats, which is a club record. The previous record was 4,992 by Carl Crawford. Longoria is also Tampa Bay's career leader in home runs, RBIs, doubles, extra base hits and total bases, and he needs three more walks to pass Ben Zobrist (542) for the franchise record in that category. . . An MRI done on Indians OF Abraham Almonte confirmed that he has a high-grade strain to his right biceps tendon. Almonte, who was placed on the disabled list Tuesday, is expected to miss three to five weeks. ... Indians DH Edwin Encarnacion, who led the American League with 127 RBI last year while with Toronto, is hitting .108 with runners in scoring position this year (4-for-37).