All 12 of last year's victorious side were in action at Austin Country Club and Alex Noren was the only player to suffer a defeat, the Swede losing 5&4 to China's Li Haotong.
Spain's Jon Rahm enjoyed the biggest winning margin with a 7&5 thrashing of Si Woo Kim, while compatriot Sergio Garcia beat Shane Lowry 4&2 and 2015 champion Rory McIlroy beat Luke List 5&4.
— The European Tour (@EuropeanTour) March 27, 2019
There were also wins for Tyrrell Hatton, Paul Casey, Ian Poulter, Henrik Stenson, Justin Rose and Thorbjorn Olesen after Fleetwood had got the ball rolling with a 3&2 win over Korea's Byeong Hun An and Molinari thrashed Satoshi Kodaira 5&4.
Fleetwood and Molinari became the first European pair to win all four of their matches together in September's Ryder Cup victory over the United States at Le Golf National, with Open champion Molinari finishing with a perfect 5-0 record.
"I've lost the first day and then won the group and I've won the first day and finished last in the group," Fleetwood told Sky Sports. "But it's great to have a good first day because once you lose you are behind the game and have to catch up.
"I had a good break around eight and nine. He had a three-putt on eight and missed a chance on nine and it kind of flipped the game around."
Molinari admitted his performance in Paris had given him extra confidence coming into an event where he has never been beyond the last 32.
"There's a bit more confidence going into match play but I have to say that's true every tournament," the Italian said. "If I compare this time last year to now obviously a lot of things have happened in between and they change you."
Casey, who successfully defended his title in the Valspar Championship last week, was one down after five holes but recovered to beat Abraham Ancer 5&3, while Stenson beat Phil Mickelson 2&1 in a match which revived memories of their duel for the 2016 Open.
Poulter looked in danger of suffering another nightmare result at the hands of Kevin Kisner when the American birdied the 15th and then holed from 60 yards for an eagle on the 16th to get back to all square.
However, Poulter birdied the last two holes to exact a small measure of revenge for his 8&6 defeat to Kisner in last year's quarter-finals. Poulter had wrongly told been told reaching the last eight would be enough for him to qualify for the Masters.
Tiger Woods, making his first appearance in the event since 2013, defeated Aaron Wise 3&1 in a scrappy contest and admitted: "Neither one of us really played that great today.
"It was a pretty emotional match because we were up and down, up and down,
there weren't a lot of holes halved and I was just kind of hanging in there. It was one of those matches where we weren't playing our best and on a golf
course like this it kind of exposed us a little bit."
Scotland's Russell Knox faces a daunting task to reach the semi-finals and qualify for the Masters after suffering a 3&1 defeat to Bryson DeChambeau, while the biggest upset saw Denmark's Lucas Bjerregaard beat world number five Justin Thomas 3&2.
The 64 players are split into 16 groups of four, with only the group winners advancing to the knockout stages at the weekend.