Mickelson, although he has been on the winning team three times, has been on the losing end all five times that he has been on the visiting side.
There were no surprises when Furyk revealed his three captain's picks in Philadelphia, with in-form 24-year-old Bryson DeChambeau, winner of the past two events on the PGA Tour, chosen along with the seasoned Woods and Mickelson.
The U.S. will defend the Cup at the Sept. 28-30 event against Europe at Le National in Paris.
Furyk said the trio were chosen with the venue in mind.
"We're heading over into foreign soil. It's going to be an interesting crowd," Furyk said.
"They are boisterous, I have a lot of respect for them, and we are looking for players that we thought would handle that situation well and would thrive, love the challenge ahead of them, and naming these three players, that's what we've done."
Furyk heaped particular praise on 14-times major champion Woods, who will now relinquish his role as a vice-captain.
"It's been great to have his help and to get a little feel of one of the best minds in the history of the game on your side helping you out," he said.
"We're all excited to see him healthy again, to see his game playing so well."
Woods, 42, will return after missing the past two Ryder Cups with a persistent back injury.
BEYOND SPECIAL
"I haven't been part of the team playing-wise for six years," Woods said. "This year to have the honour of being able to play again is beyond special.
"This team is young, full of guys in their 20s. We've got an unbelievable veteran leader in Phil who's been around the Ryder Cup since before some of these guys were born. It's a great mix of guys."
Woods has a pedestrian Ryder Cup record of 13 wins, 17 losses and three halves, though he has been stellar in singles, losing only once, on his debut in 1997.
Mickelson, 48, will be making a record 12th appearance in the event, moving past European great Nick Faldo.
"This is most likely my last chance to go over to Europe and be part of a winning team in Europe," Mickelson said, speaking of his desire to fill the void in his record.
"It's going to be a great challenge because we know how strong the European side is and how well they play at home but it's a wonderful opportunity to do something I haven't been a part of in my career.
"My excitement to be on (the team) is at an all-time high because I know how much I cherish these events."
The U.S. last won an away Ryder Cup in 1993.
Mickelson has an 18-20-7 Ryder Cup record.
Furyk also announced that David Duval, Zach Johnson and Matt Kuchar would join Davis Love III and Steve Stricker as U.S. vice-captains in Paris.
Furyk will announce his final captain's pick next Monday to complete the 12-man team.
European captain Thomas Bjorn will name his four captain's picks on Wednesday to complete the home team.
U.S. TEAM
Brooks Koepka
Justin Thomas
Dustin Johnson
Patrick Reed
Bubba Watson
Jordan Spieth
Rickie Fowler
Webb Simpson
Tiger Woods
Phil Mickelson
Bryson DeChambeau
12th player yet to be named
(Reporting by Andrew Both in Cary, North Carolina; Editing by Toby Davis and Ken Ferris)
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