"It's weird," Guyer said. "I've never seen a win like that."
Cobb (9-6) scattered two hits, walked two and struck out six over seven innings, improving to 7-0 in his last 10 starts. After Brad Boxberger worked out of an eighth-inning jam, Jake McGee got three outs for his 14th save and complete a four-hitter.
"What can I say, man?" Tampa Bay manager Joe Maddon said. "It really lived up to the billing prior to the game. Just happy we won."
Price, traded to the Tigers as part of a three-team deal on July 31, got a standing ovation while taking his warmup throws before the bottom of the first.
"That's probably as good as I've pitched in a game that went my way," Price said. "It's the least amount of hits I've ever given up."
Price, taken first overall by Tampa Bay in the 2007 draft, got the save in the Rays' Game 7 victory over Boston in the 2008 AL championship series and was the 2012 AL Cy Young Award winner.
"All the stuff leading up to the game was awesome," Cobb said. "Definitely, David deserved everything he got from the crowd and leading up to the game with all the hype. Once it became game-time, the facts are we're streaking in the wrong direction a little bit and we definitely needed the win."
Tampa Bay had lost four in a row after becoming just the fourth big league team to reach .500 in the same season after dropping 18-games under the break-even mark. The Rays lead the AL with 16 shutouts.
"Classic pitchers' duel," Tigers manager Brad Ausmus said.
Alex Avila walked and Suarez singled to open the third, but Cobb got a double-play grounder from Davis and Ian Kinsler's grounder. The Tigers also failed score in the seventh when J.D. Martinez struck out and Nick Castellanos hit a flyball with runners on second and third.
Tigers: Miguel Cabrera was the DH one day after aggravating an old right ankle injury sliding at home plate Wednesday night. "Generally speaking he's fine," Ausmus said. ... RHP Justin Verlander (sore right shoulder) threw a light bullpen with "no issues" and is scheduled to start one game in Saturday's doubleheader at Minnesota. He walked into Ausmus' office after the mound session and high-fived the manager.
Rays: C Ryan Hanigan (oblique) is scheduled to play for Class-A Charlotte Friday.
The Rays are the first team since the 1915 St. Louis Terriers of the Federal League to win and lose a one-hit game during the same season. Tampa Bay allowed one hit in a 3-2 loss to Oakland on May 21.
Tigers: Detroit heads to Minnesota for a four-game series that begins Friday night. The second AL wild-card leader trails Central Division-leading Kansas City by 1 1-2 games.
Rays: Tampa Bay opens a seven-game, two-city road trip Friday night against AL East-rival Toronto.