The move to the DL and the recall of left-handed reliever Cesar Ramos from Triple-A Durham were made before Sunday's game against Oakland.
Cobb felt what he described on Sunday as a dead feeling in his hand during his start Saturday night. He was taken to a hospital for precautionary tests. He said blood clots were among the things ruled out.
"We're going to shut him down until we feel very comfortable that he's OK," Tampa Bay manager Joe Maddon said. "We don't anticipate anything negative long term."
Maddon expects Cobb to pitch again this season.