Price acknowledged the deal Thursday on his official Twitter account, sending out his thanks to Tampa Bay fans.
wow...what a day!! Rays fans THANK YOU!! Great Chapter of my life just ended...ready to start a new one with the Tigers!! Thanks again- David Price (@DAVIDprice14) July 31, 2014
The Rays also received infield prospect Willy Adames from the Tigers.
"The question that we asked ourselves is: What gives us the best chance of winning the world championship this year?" Tigers general manager Dave Dombrowski said. "We thought adding him to our rotation at this point would give us the best chance to do that.
"Today it's great, it's nice, it's fine. I love to have David Price. If we win, then I'll be very satisfied."
The Rays said the trade was in the best interests of the franchise.
"Compared to the other possibilities, it was by far the most prudent thing that we could do," said Andrew Friedman, Tampa Bay's vice president of baseball operations. "If you look ahead to next year, it would have been really challenging to be as good as we want to be within our situation, with David taking up that meaningful of a percentage of our payroll. So everything we do is in the vein of how to be competitive in the near term and also the long term."
The Rays also aren't giving up on this season.
"It's not waving a white flag at all," Rays president Matt Silverman said. "If your team plays to the level that it can, especially considering all the head-to-head games we have within our division, we have a chance to make up the deficit and get into the playoffs. It's more difficult without David, but if the rest of the team plays up to their potential, we're going to be playing meaningful games in September."
Rays manager Joe Maddon addressed the trade on Twitter, writing:
No fun losing a David Price. Hate it. But we have to do what is best for our group. Continue to move it forward and that's what we've done.- Joe Maddon (@RaysJoeMaddon) July 31, 2014
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 with one out in the top of the seventh and was seen hugging several teammates before exiting into the clubhouse.
"It's kind of a weird moment right there taking that last jog off the field knowing that's pretty much the last time I will be running off in a Tigers uniform," said Jackson, who is hitting .273 and has been Detroit's regular center fielder throughout its recent run of division titles.
"I really didn't know what to do because I've never been in that situation where you're taken out of a game in the middle of an inning. I knew what was going on, I was just in a daze a little bit and was trying to let everything play out."
Smyly started the game but lasted just five innings, allowing four runs and 11 hits. The 25-year-old is 6-9 with a 3.93 ERA this season. He said he found out about the trade on social media, and Dombrowski came in and told him soon thereafter.
"I don't really have anything to say. It was unexpected. I didn't see it coming," Smyly said.
"I didn't want to leave here. These guys are awesome. We have a great team, a winning team, so it's hard to say goodbye, but it's part of baseball. It happens to everybody."
Price, the AL Cy Young winner in 2012, joins a Tigers rotation that already features Justin Verlander (the 2011 AL Cy Young winner), Max Scherzer (the 2013 AL Cy Young winner), Anibal Sanchez and Rick Porcello. The Tigers are the first team with three pitchers on the staff who all have won Cy Young Awards over the previous three seasons, according to STATS. Detroit entered Thursday's game with a five-game lead atop the AL Central standings.
"David Price comes over and fills out an extremely strong rotation," Verlander said.
"It's tough to see somebody go but you look at the positives of bringing over a pitcher with quality stuff. I actually know David a little bit, and he's a great guy. Smyly and Jackson are good locker room guys, [and] David is a good locker room guy as well. He will fit into this locker room quite nicely."
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 past two postseasons, with the Tigers winning both matchups.
Verlander addressed comments he made earlier this month in which he said the Athletics made moves in reaction to the Tigers.
"I think that was taken out of context a bit," he said. "Obviously the A's are making moves to win a World Series, not to get past us, which is what happened the last couple of years. We were the team they didn't quite get by. I don't think they're sitting there saying, 'We need to make this move to beat the Tigers,' and I don't think vice versa. They made their big moves; we made our move. This might have been something on the horizon for a while for us. I don't think it's one of those things where, 'OK, this team made a move, now we need to counteract that.'
"But I can tell you it was an exciting day of trade-deadline baseball. This was the most interesting day I have been a part of. I don't think I have been a part of having to get somebody off the field because it's almost 4 o'clock and saying, 'All right, we have to go get him. We can't wait any longer.' It's crazy."
The Tigers have won three straight division titles, but they're still without a World Series championship since 1984. Last season's team lost to Boston in the AL Championship Series.
Price's departure became a clear possibility when the Rays fell way behind in the AL East race earlier this season. Tampa Bay has played much better over the past month and entered Thursday only 5 ½ games behind a wild-card spot -- but the Rays decided to deal their ace anyway.
Tampa Bay signed Price to a $14 million deal for this season, avoiding arbitration.
Rays teammate Evan Longoria tweeted that he was sad to see Price go.
All I can say is...Thank you @DAVIDprice14 for all the wins and memories. Your character is unmatched.- Evan Longoria (@Evan3Longoria) July 31, 2014
Franklin, 23, hit .294 in 75 games this season with Triple-A Tacoma. He appeared in 17 games with the Mariners, hitting .128.
With a lineup anchored by Miguel Cabrera, Victor Martinez and Ian Kinsler -- and that remarkable rotation -- Detroit has remained in first place for almost all of 2014. The Tigers tried to shore up one trouble spot -- the bullpen -- by trading for Joakim Soria last week.
Now Detroit has emerged with Price in another bold move aimed at bringing a championship to the Motor City.
"I don't know if we're going to win by any means -- there's other good clubs," Dombrowski said. "But getting the opportunity to add a guy like David Price doesn't come along very often."
ESPN.com's Jayson Stark, ESPNChicago.com's Doug Padilla and The Associated Press contributed to this report.