The U.S. won three of five foursomes matches, and halved another, to enjoy a 3-1/2 to 1-1/2 advantage in strong and gusty winds that made club selection difficult.
It could have been even worse for the International team as American veteran Phil Mickelson missed a six-foot putt at the final hole as he and partner Kevin Kisner halved with Australians Jason Day and Marc Leishman.
The International team, comprised of players from the rest of the world excluding Europe, have now lost the opening session six consecutive times at the biennial event.
"We're one point better off then we were two years (ago)," Price said. "It was a brutal day to play golf. We got behind the eight-ball early with a few of our teams being two-down, three-down at the turn.
"But the guys rallied back great. Even though we're two points behind, it was one of our strongest starts in foursomes on Thursday.
"So the team is not discouraged at all. We have a resilient team. They are raring to go tomorrow."
The American pair Justin Thomas and Rickie Fowler set the tone in the first match, thrashing Hideki Matsuyama and Charl Schwartzel 6&4.
The Americans also had an emphatic victory in match three as Jordan Spieth and Patrick Reed thumped South Korean Kim Si-woo and Argentine Emiliano Grillo 5&4.
The International team suffered another disappointment when Australian Adam Scott and Jhonattan Vegas of Venezuela were edged one-up by Dustin Johnson and Matt Kuchar.
The result leaves Scott with more foursomes defeats, nine, than any player in Presidents Cup history. He is 4-9-1 in the alternate shot format. Scott also joins South African Ernie Els with a record 18 losses over all formats.
STELLAR PARTNERSHIP
The International team did manage to salvage something from the other two matches as South Africans Louis Oosthuizen and Branden Grace continued their stellar partnership with a 3&1 win over Brooks Koepka and Daniel Berger.
Oosthuizen and Grace paired up to win four matches at the 2015 event.
In the final match, Day and Leishman recovered from a big early deficit to take the lead against Mickelson and Kisner.
But the Australians opened the door with bogeys at the final two holes, and were fortunate to escape with 1/2 a point.
U.S. captain Sticker was pleased with the day overall.
"There's still a long, long ways to go. But we very much liked the day and the way it started," he said.
The International team has only one overall victory (1998) and one tie (2003) in 11 stagings of the event.
The Cup continues on Friday with five four-ball (better ball) matches.
Price will rest rookies Kim and Grillo, while U.S. captain Steve Stricker opted to leave out Matt Kuchar and Daniel Berger.
Four-ball matches at Presidents Cup on Friday (U.S. named first):
11.35 a.m. ET (1535 GMT) - Jordan Spieth/Patrick Reed v Hideki Matsuyama/Adam Hadwin
11.50 a.m. ET - Rickie Fowler/Justin Thomas v Louis Oosthuizen/Branden Grace
12.05 p.m. ET - Phil Mickelson/Kevin Kisner v Jason Day/Marc Leishman
12.20 p.m ET - Kevin Chappell/Charley Hoffman v Charl Schwartzel/Anirban Lahiri
12.35 p.m. ET - Dustin Johnson/Brooks Koepka v Adam Scott/Jhonattan Vegas
(Reporting by Andrew Both, editing by Pritha Sarkar and Ken Ferris)
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