The Rays also received infield prospect Willy Adames from the Tigers, according to reports.
The deal was completed just ahead of Thursday's 4 p.m. ET deadline for non-waiver trades.
Jackson was pulled from Detroit's game Thursday afternoon against the Chicago White Sox and was seen hugging several teammates before exiting into the clubhouse. Smyly started the game but lasted just five innings, allowing four runs and 11 hits.
Price, the American League Cy Young award winner in 2012, joins a Tigers rotation that already features Justin Verlander, Max Scherzer, Anibal Sanchez and Rick Porcello. Detroit entered Thursday's game with a five-game lead atop the AL Central standings.
Price is eligible to become a free agent after the 2015 season and could serve as a short-term replacement in Detroit for Scherzer, who will hit free agency this winter after declining the Tigers' reported six-year, $144 million offer.
But at least for the remainder of the 2014 season, the Tigers will have a star-studded rotation to rival that of the Oakland Athletics, who acquired Jon Lester from the Boston Red Sox in Thursday's other major trade.
Detroit and Oakland have played each other in each of the last two postseasons, with the Tigers winning both matchups.