Dallas quarterback Tony Romo completed 22-of-29 passes for 307 yards and three touchdowns, with receiver Kevin Ogletree caught eight passes for 114 yards and a pair of touchdowns, while DeMarco Murray rushed for 129 yards on 19 carries.
The Cowboys built a 24-10 lead with less than six minutes to play after a leaping catch by Miles Austin turned into a 34-yard touchdown.
New York reduced the deficit with 2:36 remaining in the fourth quarter when Eli Manning and Martellus Bennett connected, but they could make no further inroads.
"A huge emphasis for us was big plays," said Ogletree, who grew up in the New York borough of Queens.
"I don't want the focus to be on me too much, but I am very, very humbled and appreciative of how we played today."
The visitors took a 7-3 lead into the intermission and built a 14-3 advantage following the kickoff for the second half when Ogletree, an undrafted receiver out of Virginia in his fourth season, hauled in a 40-yard touchdown pass.
New York responded with a 10-yard touchdown run by Ahmad Bradshaw, set up by a 39-yard pass to Domenik Hixon, to make it 14-10.A 33-yard field goal by Dan Bailey made it 17-10 for Dallas before the teams traded fourth-quarter touchdowns to complete the scoring.
Dallas outgained the Giants 433 yards to 269, and though New York's vaunted pass rush registered a pair of sacks, Cowboys quarterback Romo scrambled out of trouble to make several big completions.
Manning, the Most Valuable Player of New York's Super Bowl win over New England last season, faced heavy pressure from the Dallas rush and was sacked three times.
He completed 21-of-32 passes for 213 yards and a touchdown.Both teams lost starting players to injuries during the game, with Dallas center Phil Costa sidelined with back spasms after his third play from scrimmage and New York having their depleted defensive secondary hit further by a hamstring injury to Michael Coe.
The game was marked by sloppy play and a slew of penalty flags thrown by the replacement officials, employed by the league during a contract dispute with the regular crews.
"This is a hostile environment, a tough place to play," said Dallas coach Jason Garrett.
"When you play the Super Bowl champs at their place on opening night, it's a difficult task. We viewed it as a challenge."
Giants coach Tom Coughlin refused to make excuses for his players after the defeat.
"Humble pie is basically what it is," he said after the game.
"Hopefully the competitive nature of our guys, they will come out swinging (next time)."
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