Romo rallied the Cowboys from a 10-point halftime deficit with touchdown throws of 45 yards to Cole Beasley and 31 yards to a wide-open Dez Bryant.
The second touchdown pass came moments after Barry Church intercepted Eli Manning's pass in the shadow of the end zone and returned it 45 yards to the Dallas 48.
Until then, the game belonged to Beckham and his spectacular 43-yard touchdown catch.
The rookie receiver gained position on cornerback Brandon Carr along the right sideline, left his feet at about the 3-yard line and reached as far back as he could with his right hand and pulled in the heave from Manning :08 into the second quarter for a 14-3 lead.
The catch set the crowd at MetLife Stadium abuzz and set the internet ablaze:
"Man I just witnessed the greatest catch ever possibly by Odell Beckham Jr! WOW!!!!" LeBron James tweeted.
It was as good or even better than the helmet grab David Tyree made to help the Giants beat the then undefeated Patriots in the Super Bowl in 2008.
It also was Beckham's second TD of the game. He scored on a routine 3-yarder on New York's opening possession to cap an 80-yard drive. It marked the second straight game the Giants have scored a touchdown on their opening drive after failing to do so in their previous 20 games.
Fellow rookie Andre Williams scored on a 3-yard touchdown run late in the quarter to push the lead to 21-10.
Manning, who threw five interceptions last week, was unscathed until he overthrew a wide-open Preston Parker around the Dallas 10 and Church came up with the tipped ball.
Romo helped Dallas shake off Beckham's grab, leading a 77-yard drive on the ensuing possession. It was capped by a two-handed catch by tight end Jason Witten on a 4-yard shovel pass.
Dan Bailey hit a 38-yard field goal to get Dallas on the scoreboard after Beckham's first touchdown.
New York is trying to snap a five-game losing streak and keep its slim playoff hopes alive. Dallas is looking for a win to tie Philadelphia (8-3) in the NFC East.