The 38-year-old Woods, making a comeback after a four-month injury lay-off, stumbled to five bogeys and a double bogey.
Gangjee battled tooth-and-nail after a poor front nine 37 but was in contention again following an eagle on 14.
He also plans to defend his title in the Australian Open, which would be his final tournament of the year.
Randhawa began with a bogey but went on to reel off 7 birdies to lead Angelo Que of Philippines by 1 shot.
Rory McIlroy set the tone with a 5&4 demolition of Rickie Fowler & Graeme McDowell soon added a 2nd point.
Europe roar back to lead by 2 after the United States had gone ahead at half-way mark.
McIlroy has already enjoyed a superb 2014 campaign, clinching two major titles among four wins worldwide.
Rory’s shot on the 14th hole clattered into a tree & landed into the front right pocket of a man’s shorts.
The coach controversially redesigned Woods’s swing but the American great has failed to win a major.
World No.1 McIlory trails the leading trio in the clubhouse by eight strokes after first round of The Barclays.
Tiger Woods has appeared on seven U.S. Ryder Cup teams between 1997 and 2012 but missed the 2008 edition.
At 25, he is the third-youngest golfer to win four majors, behind Tiger Woods and Jack Nicklaus.
Phil Mickelson narrowly missed out on a sixth major victory at the PGA Championship; made his way in to the US team.
Aged just 25, the Northern Irish world number one became the fourth youngest player to land four majors.
McIlroy, who won 2012 PGA Championship at Kiawah, became the fourth youngest player to land four majors.