Dallas quarterback Tony Romo completed 22-of-29 passes for 307 yards and three touchdowns.
Unheralded Kevin Ogletree, who had no touchdown receptions in his first three seasons in the league, caught eight passes for 114 yards and a pair of touchdowns, while DeMarco Murray rushed for 129 yards on 19 carries.
"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."
The opening night program began with NFL Commissioner Roger Goodell ushering in the new season and saluting the Giants to the roars of more than 80,000 fans, but before long it was the Cowboys who were dominating on the ground and in the air.
"Humble pie is basically what it is," said Giants coach Tom Coughlin. "Hopefully the competitive nature of our guys, they will come out swinging (next time).
"You're not going to get excuses from us tonight. They beat us."
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 before the Giants closed in with a 79-yard scoring drive.
Dallas ran the clock out after Romo connected with Ogletree for 15 yards to convert a third-and-12 with about two minutes left in the game, as the Cowboys avenged two late season losses to New York that kept them out of the playoffs.
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.
Giants quarterback Eli 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.
The rest of the NFL's Week One schedule will be played on Sunday and Monday.