Couples chose former major winner Webb Simpson and rookie sensation Jordan Spieth as his wild cards for the US team. Price went for fellow Zimbabwean Brendon de Jonge and Australian Marc Leishman to round out his 12-man International lineup.
Spieth, Leishman and de Jonge will be making their first appearances in the biennial team competition while Simpson is set for his second stint with the US after making his Cup debut in Australia in 2011.
Simpson, the 2012 US Open champion, was widely expected to be included for the Ryder Cup-style competition after spending most of the past two years in the automatic top 10 in the US points standings before finishing up in 11th.
"I am so excited," said Simpson. "Playing in Australia (at the 2011 Presidents Cup) and coming out victorious was just so much fun. I can't think of anybody I would rather play with than (fellow American) Bill Haas, but quite honestly, I'm comfortable playing with anybody."
Red-hot US Tour rookie Spieth was a much bolder selection but had captured Couples' attention with a sizzling run of form over the past two months.
The 20-year-old won his maiden US Tour title at the John Deere Classic in July, lost out in a playoff for last month's Wyndham Championship and rocketed into a tie for fourth at the Deutsche Bank Championship on Monday after closing with a 62.
"To pick Jordan was the right move and he is having an incredible year," Couples said after announcing his two picks on Golf Channel. "He is one of our top players, even though he is only 20. I'm excited to have a young kid on the team.
"I left Keegan (Bradley) out two years ago, and that was a killer decision. He has become one of the best young players in the game. It's a team game, and everybody I talked to on the team wanted Spieth. He definitely deserves it."
Spieth said: "It's a different style (in matchplay golf). You get to go out there, and get a little more fired up. You feel like you have the support of another guy right next to you. It just brings a level of comfort to me. I grew up playing a lot of team sports, really enjoyed that."
Couples said he had always planned to pick Simpson, though he was bitterly disappointed to have no room on his team for Jim Furyk, a veteran of seven Presidents Cups.
"I haven't talked to him (Furyk) yet," Couples added. "I told him last night that he wasn't on the team through a text and would talk to him today just because I just couldn't handle it, to be honest with you.
"The two guys that I wanted dearly to be on the team were Steve Stricker and Jim Furyk ... that's not to say that I didn't want any of the other 10 guys. I took the easy way out. Jordan deserves to be on the team and Webb is a class act."
Burly Zimbabwean de Jonge, though yet to win on the US Tour, has established a reputation as a 'birdie machine' on the US circuit and gained Price's nod after a consistent 2013 campaign capped by a strong start to the FedExCup playoffs.
"This was my primary goal starting out last year to make this team, and especially to play for (fellow Zimbabwean) Nick (Price)," said de Jonge. "Nick has been everything to us in junior golf growing up in Zimbabwe and now I have the opportunity to play for him, it's definitely a dream come true."
Leishman, whose only US Tour victory came at the 2012 Travelers Championship, has recorded four-top-10s this season and effectively sealed his Cup debut by finishing 16th at the Deutsche Bank Championship.
""It's exciting for us to have three Australians in the team and for me personally it's huge," said Leishman. "It's been a big goal of mine for the last few years to try and get on this team. Having to rely on a captain's pick probably wasn't ideal but to be lucky enough for (Internationals captain) Nick (Price) to pick me was awesome.
The biggest headache for Price was having to leave out South African Tim Clark, a relatively short but accurate hitter off the tee who is known for his matchplay prowess and has played in three Presidents Cups.
"Yesterday was probably one of the toughest days of my life," said Price. "I had it down to three guys - Marc, Brendon and Tim Clark. To try and separate the three of them and only pick two was an exceptionally difficult task.
"You had experience on one hand with Tim, and then you had youth in the two rookies in Marc and Brendon. When push came to shove, it came down to who I felt is going to play Muirfield the best.
"Muirfield Village is a specialist golf course and both Marc and Brendon cut the ball and fade the ball, and they hit the ball pretty far."
The US have dominated the Presidents Cup, winning seven times in nine editions, including a 19-15 victory in the most recent edition at Royal Melbourne in Australia two years ago.
The only success for the Internationals came in 1998 when the event was first staged in Melbourne. In 2003, the two teams battled to a 17-17 draw in South Africa.
US team: Tiger Woods, Brandt Snedeker, Phil Mickelson, Matt Kuchar, Keegan Bradley, Jason Dufner, Steve Stricker, Bill Haas, Hunter Mahan, Zach Johnson, Webb Simpson, Jordan Spieth.
International team: Adam Scott (Australia), Charl Schwartzel (South Africa), Jason Day (Australia), Ernie Els (South Africa), Louis Oosthuizen (South Africa), Hideki Matsuyama (Japan), Branden Grace (South Africa), Graham DeLaet (Canada), Richard Sterne (South Africa), Angel Cabrera (Argentina), Marc Leishman (Australia) and Brendon de Jonge (Zimbabwe).