Volquez struck out nine but allowed a career-high eight walks before departing with a thumb blister after 4 1/3 innings, and Tampa Bay took advantage of his historically uneven outing to beat Miami 3-1 Tuesday.
Cobb (2-2) pitched six scoreless innings despite a small blister on his middle finger and earned his first win since the first week of the season.
Logan Morrison hit his sixth home run and was one of only five batters to put the ball in play against Volquez (0-4). But the Rays still managed three runs off the right-hander.
"It was hard for me to have a good grip on the ball and command my fastball," Volquez said. "I had like 20 different grips today. It's really hard."
Volquez threw just 42 of 91 pitches for strikes and became the first starter to walk at least eight batters and strike out at least nine while pitching less than five innings since 1900, the Marlins said, citing information from the Elias Sports Bureau.
"I'm in the book now," Volquez said with a smile. "It's crazy."
The blister on the inside of Volquez's thumb bled and grew larger than a pencil erasure. He said it's worse than previous blisters he has had in the same spot, and he's not sure whether he'll have to skip a start.
"I've never missed a start with a blister, but it's really bad right now," he said. "We've got to wait a couple of days to see."
Cobb's blister developed before the game and drew the attention of manager Kevin Cash and the Rays trainer in the first inning, when they visited the mound.
"They thought I was grimacing," Cobb said, "but that was just my reaction to the two poor pitches that I threw. It was a little bit tender when I was gripping the ball. But it didn't get too hot. It was fine to keep pitching."
Three relievers finished, and Alex Colome earned his seventh save by striking out the side in the ninth to complete a six-hitter.
The Rays climbed to .500, and improved to 5-10 on the road. The Marlins lost for the sixth time in their past seven games and have totaled three runs and 10 hits in the first two games of the series.
"One of those bumps in the road. No energy," slugger Giancarlo Stanton said. "We've got to pick up the pace -- more determination to get out of what we're in."
Rays hitters totaled eight hits, 15 strikeouts and 11 walks, including four to Brad Miller. Miami pitchers issued at least one walk in each of the first eight innings and have given up 19 bases on balls in the first two games of the series.
Morrison's homer made it 1-0 in the third. After the Rays loaded the bases with three walks in the fifth, Tim Beckham hit a sacrifice fly, and Derek Norris singled home another run.
Stanton doubled home Miami's run in the eighth.
Morrison spent his first four seasons with the Marlins and acknowledged it was satisfying to homer against his former team.
"Yeah, absolutely," he said. "I'm not going to forget."
Given a rare chance to hit, Cobb fouled a ball off his back foot, adding to his discomforts.
"Par for the course for me," he said. "I tried to hack at a sinker, and it somehow hits me in the back foot. It was a standard pitcher moment trying to hit."
Colby Rasmus made his Rays debut in the seventh and walked pinch hitting. He was activated from the DL before the game following his recovery from hip surgery.
Rays 3B Evan Longoria missed his second game in a row because of plantar fasciitis in his left foot. The Rays are optimistic he'll play Wednesday.
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