The deal was first reported by ESPN The Magazine's Buster Olney.
"Honestly, I was shocked by this, but that's baseball," Rodriguez said. "Now, I'm just looking forward to playing for the Yankees and maybe I will play against Detroit again in the playoffs."
After New York's 13-3 rout of the Orioles on Wednesday, Yankees manager Joe Girardi expressed his satisfaction with the deal.
"He's hitting over .290 I believe and he's been hot so we're looking forward to having him," he said.
Rodriguez, who is in the final year of the contract he signed with the Tigers in 2004 that will ultimately pay him $50 million, waived his no-trade clause to join the Yankees.
"Pudge has always wanted to play in New York and Detroit felt it needed a relief pitcher, so both sides were able to get what they wanted to some degree," Rodriguez's agent Scott Boras told The Associated Press.
Rodriguez is batting .295 this season with five home runs and 32 RBIs in 82 games. In 18 seasons with the Rangers, Marlins and Tigers, he's hit .302 with 293 homers and 1,214 RBIs.
"It's going to be very exciting. They are in the race and I'm ready to do my job," Rodriguez said.
A 14-time All-Star, Rodriguez won the American League MVP award in 1999 and for years was known as the best defensive catcher in the game. He earned Gold Gloves in 13 seasons, a record for a catcher.
Willie Mays, Roberto Clemente and Rodriguez are the only three players in baseball history with at least 12 Gold Gloves and a career .300 average or better. Rodriguez has caught 2,142 games, trailing only Carlton Fisk (2,226) and Bob Boone (2,225) in terms of longevity behind the plate.
"He's been a great player for a long time and I remember the battles we had in Texas. He's been a world champion," said Girardi, who was a member of the Yankees during playoff battles with the Rangers in the '90s.
In his lone season in Florida in 2003, Rodriguez helped lead the Marlins to a World Series victory over the Yankees.
The Yankees expect Rodriguez to join the team on Thursday, when they start a four-game series against the Los Angeles Angels.
Rodriguez has said he wants to play next season and beyond, but for now is focusing solely on helping the Yankees catch Tampa Bay and Boston in the AL East.
"It's going to be very exciting," he said. "They are in the race and I'm ready to do my job."
Farnsworth is 1-2 with a 3.65 ERA this season. Farnsworth pitched for the Tigers for the first four months of 2005, compiling a 2.32 ERA in 46 games before he was traded to the Braves at the trading deadline.
"Another team wants me. That's the way I look at it," he said. "
They [the Tigers] know my reputation. They know what type of person I am."
The departure of Farnsworth creates a void in the eighth inning for the Yankees.
When Joba Chamberlain moved into the starting rotation and gave up that role, Farnsworth eventually stepped in and had been effective.
"He's been throwing the ball great and it's great to see the things he's done to make himself better, but it's a business and you have to understand that," Chamberlain said.
Yankees general manager Brian Cashman summed up the risk: "We are robbing Peter to play Paul so we hope it works out."
Cashman went on to say that though he had heard Rodriguez might be on the trade block earlier in the season, discussion between the teams started on Wednesday.
"Somewhere sixth, seventh inning I told Joey [Girardi] I don't know where this is going but don't use Farnsworth," Cashman said.
As for who takes over in the eighth, Cashman said the acquisition of Damaso Marte in the trade that brought outfielder Xavier Nady to the Yankees from Pittsburgh helps, as does the development of Edwar Ramirez. In addition, injured starter Phil Hughes has made a rehab start and could be close to returning.
The deal finally came down to the Yankees' lack of depth at catcher.
"Pudge is an everyday guy," Cashman said. "It comes down to balancing this thing out."
Tigers president Dave Dombrowski said the team didn't want to trade Rodriguez but had bigger needs in the bullpen.
"First let me say that Pudge has done an outstanding job for us," Dombrowski told reporters before Detroit played the Cleveland Indians. "I'm sure he was very surprised by this, but for him this is a good situation. He's going to New York City, and they are in contention."
Dombrowski said it would be up to manager Jim Leyland to decide where to slot Farnsworth in the bullpen. Fernando Rodney, who has one career save, is the new closer for the Tigers after Todd Jones lost the role. Rodney came in Tuesday night in a nonsave situation and gave up a run in the ninth.
The Tigers' GM also said Brandon Inge would take over at catcher.
"We think Brandon Inge can do a fine job at catcher and at the same time we've done some repair to our bullpen," Dombrowski said.
Inge said he was excited to be getting a chance to start behind the plate on an everyday basis for the first time since 2003, when he started 98 times at catcher. Since then, he's gotten most of his starts at third base.
"I wanna play every day no matter what position, so I'm pumped," he said. "Did I learn a lot from Pudge? He's a future Hall of Famer, so watching him makes everybody better. I'm not Pudge, I'll just go out and do what I can."
Tigers slugger Gary Sheffield said the trade made sense to both sides.
"This is a surprise to people if they don't know the situations of the players," he said. "Both teams get better by this, and that's what a trade is supposed to be all about."
Rodriguez appeared in the Tigers' completely empty dugout hours before Wednesday's game and waved over to Indians players, who were taking batting practice. He then disappeared back into the tunnel.
Rodriguez will be remembered in Detroit for being the star who sparked the franchise's turnaround when nobody wanted to be a Tiger -- at any price.
He signed a $40 million, four-year deal with the Tigers after they lost an AL-record 119 games in 2003. He had just helped Florida win the World Series, earning the MVP award in the NLCS against the Chicago Cubs.
In 2006, Rodriguez and the Tigers reached the World Series.
The player known as Pudge was an All-Star in each of his first four seasons with the Tigers. He also won three more Gold Gloves.
Posada, a five-time All-Star catcher, announced he would have season-ending surgery on his right shoulder on Monday. He had the surgery on Wednesday, and the rehab time is about six months. Posada has struggled with shoulder pain for most of the season, limiting his ability to throw out baserunners and cutting down his playing time behind the plate.
On the disabled list for the second time this year, Posada was trying to rehabilitate his shoulder enough to come back as a designated hitter or first baseman this season. But after the Yankees acquired Nady in a trade with Pittsburgh on Saturday, they determined it was best for Posada to have surgery now, Cashman said Monday.
Later in the day, the Yankees traded reliever LaTroy Hawkins to
Houston for Class-A infielder Matt Cusick. New York designated
Hawkins for assignment Saturday after acquiring Nady and Marte.
Buster Olney covers baseball for ESPN The Magazine. Information from The Associated Press was used in this report.