Ben Zobrist reached on a throwing error by shortstop Eugenio Suarez scored on Guyer's hit. Price then retired his final 23 batters, nine on strikeouts, to finish an eight-inning complete game.
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.
Kevin Kiermaier, inserted as a defensive replacement in right field, made diving catch on Rajai Davis' flare with a runner on second and one out in the eighth.
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. The left-hander, 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.
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.