Romo threw for 307 yards in subduing the Giants, whose regular season-ending victory over the Cowboys a year ago launched them to the Super Bowl title.
Ogletree caught a 10-yard scoring pass with 61 seconds left in the first half to give Dallas a 7-3 lead at the break and he ended an 80-yard march in the third quarter with a 40-yard touchdown reception.
Romo then found Miles Austin with a 34-yard touchdown throw in the fourth period to give the Cowboys a 24-10 advantage and Dallas held on to defeat its division rivals for just the second time in their last seven meetings.
New York closed within seven points when Eli Manning threw a 9-yard touchdown pass to former Cowboy Martellus Bennet with 2:36 to play and the Giants chose to kickoff deep. New York never got the ball back.
Facing a third-and-2 from its own 34 after the kickoff, Dallas appeared to have picked up the first down only to have a holding penalty called on tight end Jason Witten.
Given an opportunity to stop Dallas and regain the ball with plenty of time left to tie the game, the Giants then gave up a 13-yard pass from Romo to Ogletree for a first down that clinched the victory.
Romo completed 22-of-29 passes and his only mistake was a second-quarter interception by Michael Boley, who returned it 51 yards to the Cowboys' 2-yard line.
New York, however, eventually had to settle for a field goal that turned out to be its only points of the first half.
Ogletree caught eight passes for 114 yards and DeMarco Murray, who did not play in the final game against the Giants last year due to injury, ran for 131 yards on 20 carries. His 48-yard run set up a Dallas field goal in the third quarter that put the Cowboys in front 17-10.
Manning was 21-of-32 for 213 yards with a touchdown and no interceptions. Ahmad Bradshaw picked up 78 yards on 17 carries.
Dallas had a 433-269 advantage in total yardage.
There will be 13 games on the NFL schedule Sunday and the opening weekend of the campaign will conclude with a pair of Monday night games sending Cincinnati to Baltimore and San Diego to Oakland.