Austin Jackson was pulled off the field in the seventh inning of Thursday's game after the trade was finalized. Jackson was traded to the Seattle Mariners in the deal.
Drew Smyly was sent to Tampa Bay in the deal. Seattle also sent Nick Franklin to Tampa Bay.
Tigers prospect Willy Adames has also been sent in the deal. The highly-regarded shortstop will go to Tampa Bay.
Price owns an 11-8 record and 3.11 ERA this season. He is under contract through next season.
Adding Price is the second move of the month for the Tigers, who also acquired dominant bullpen pitcher Joakim Soria on July 23.