The Giants (0-1) drove 13 plays and 79 yards downfield after the Cowboys brought it within 23-20 on a Witten touchdown, but on third-and-goal from the 1 and the Cowboys out of timeouts, opted to throw on third down.
Eli Manning's pass was incomplete to stop the clock and set up Josh Brown's fourth field goal of the game to make it 26-20 with 1:37 left.
The Cowboys (1-0) then drove 72 yards downfield in 1 minute, 27 seconds, capped by Romo's 11-yard touchdown pass to Witten over the middle and Dan Bailey's extra point to give the Cowboys a one-point lead.
New York's laterals at the end of regulation failed.
Romo threw 356 yards, three touchdowns and two interceptions on 36-of-45 passing, while Manning threw 171 yards on 19-of-35 passing.
Odell Beckham Jr. caught five passes for 44 yards, Rashad Jennings had 13 carries for 52 yards and a touchdown, and Shane Vereen totaled 41 yards.
Witten had eight catches for 60 yards and two touchdowns and Dez Bryant, who left the game with an injury, was limited to five catches for 48 yards.
Trumaine McBride forced two turnovers that led to two touchdowns. With the Giants up 16-13 in the fourth, Romo's pass to Devin Street popped in the air and was intercepted by McBride, who returned the ball to the 1 to set up Jenning's touchdown that made it 23-13 with 8:01 to play.
Dallas brought it within seven on Romo's one-yard touchdown pass to Witten with 5:08 left.
The Giants responded and drove downfield to the 1. The Cowboys were out of timeouts after second down, and the Giants opted to throw the ball on third down. Manning's pass sailed high in the back of the end zone to stop the clock and set up fourth down.
The Giants forced three turnovers in the game, two of which coming toward the end of the first half.
With 1:12 left in the second quarter, McBride forced a Cole Beasley fumble after a first down. Dominique Rodgers-Cromartie picked the ball up and zig-zagged his way downfield for a 58-yard return for a touchdown to give the Giants a 10-6 lead.
On Dallas' next play from scrimmage, Witten tipped Romo's pass in the air to MLB Uani Unga for the interception. Brown's 40-yard field goal gave the Giants a 13-6 halftime lead.
The Giants host the Atlanta Falcons in their second game of the season Sept. 20 at 1 p.m. at MetLife Stadium.
A frustrating a loss as the Giants have had in years. They stole the game by converting three Dallas turnovers into 17 points and then somehow gave it back in the last minute and a half.
The Cowboys were dead meat, but for some reason the Giants, who haven't made a habit of winning lately, lost their heads and let themselves get beat.
Both coordinators and the head coach deserve the blame as does the QB. They had the clock, the ball and the score on their side. A run call on third down on the final drive would have eaten enough up enough time to prevent the Cowboys from mounting the winning TD drive.But instead, the Giants had Manning rolling out and throwing to no one. And, foolishly, he did. The incomplete pass stopped the clock and the Giants settled for a field goal, leaving the game for the defense to win. And they didn't.
On the ensuing drive, the defense was as invisible as a defense could be. Romo took the Giants to school, driving down the length of the field uncontested, finishing the Giants off with a touchdown pass to Witten (who else?) with just :07 remaining on the clock.
There's no defending this loss, no matter how die hard of a fan you are. You might be able to find some solace in the fact that they were competitive, but you can negate that by pointing out the poor coaching at the end of the game.