It was the seventh straight win for the Rays, who maintained the lead in the race for the two American League wild card spots with their 90th victory of the season.
Cobb (11-3) allowed three hits and struck out four in winning his third start in a row. He gave up a single in the first and did not allow another hit until Curtis Granderson doubled leading off the eighth.
Ivan Nova (9-6) took the loss, allowing two runs on eight hits in seven innings.
One night after the Yankees were eliminated from the playoff chase, Alex Rodriguez remained on the bench and reportedly will not play in New York's final series of the season against Houston.
Rodriguez has played in 44 games this season while appealing a four-month suspension.