The Cowboys limited the Giants to 233 total yards, paving the way for Dallas' 19-3 victory on Sunday night at AT&T Stadium.
Elliott, playing after an injunction filed against the NFL allowed him to start the season despite a six-game suspension, rushed for 104 yards on 24 carries.
The Cowboys didn't score in the second half until 1:55 remaining in the fourth quarter. Dan Bailey booted his fourth field goal, from 36 yards, for the game's final points.
The lack of second-half offense didn't matter after Dallas controlled the first half on the way to a 16-0 halftime lead.
The Giants opened the second half with a 68-yard drive that more than doubled their total yards to that point. New York reached the Dallas 4-yard line before Charles Tapper sacked Eli Manning for a 9-yard loss, causing the Giants to settle for Aldrick Rosas' 25-yard field goal.
That ended up being New York's most successful drive of the game.
Anthony Brown intercepted a Manning pass at the Cowboys 48-yard line midway through the fourth quarter to set up Dallas' final scoring drive.
Jason Witten had a productive first half as the Cowboys tight end became the team's career receiving yards leader and scored Dallas' first touchdown of the season.
Quarterback Dak Prescott tossed to Witten for an 11-yard gain in the second quarter that brought Witten's career total to 11,890 receiving yards, passing Michael Irvin's previous team record of 11,888.
Witten then capped a touchdown drive late in the first half. Prescott hit the tight end on a slant pass for a 12-yard score that put Dallas ahead 13-0 with 1:41 left in the second quarter.
Witten finished the first half with six receptions for 49 yards, and he ended the game with seven catches for 59 yards.
The Dallas defense dominated the first half, holding New York to 50 yards of total offense and just two first downs.
Without Odell Beckham Jr., who sustained a left ankle injury in the preseason and missed the Sunday game, Manning passed for 33 yards in the first half.
Dallas forced one final three-and-out before the break that allowed the Cowboys' offense to move in for Bailey's third field goal, this one from 42 yards. Bailey also made kicks from 21 and 48 yards in the half.
Prescott completed 24 of 39 passes for 268 yards, one touchdown and no interceptions. Manning was 29 of 378 for 220 yards and one interception.
NOTES: The suspense as to whether WR Odell Beckham Jr. would be healthy enough to play in the season opener ended a few hours before game time when Beckham was listed as inactive. ... Giants QB Eli Manning made his 200th consecutive start. He is only the third quarterback to reach 200, joining Brett Favre (297) and Peyton Manning (208). ... After it appeared that RB Darren McFadden would step into the hole left by Ezekiel Elliott's apparent suspension, Elliott's injunction against the NFL meant that the Cowboys listed McFadden as inactive for the season opener. RBs Alfred Morris and Rod Smith served as Elliott's backups. ... Cowboys WR Terrance Williams sustained an ankle injury on the first offensive series of the game and was listed as questionable to return. However, he re-entered the game on the last play of the first quarter and caught a 15-yard pass to convert a third-and-12.