The victory was the first for the Giants in seven Monday night meetings with the Cowboys.
It also atoned for a 31-7 thrashing on a Monday night by Dallas last season, when Sanders returned a punt and an interception for a touchdown.
This was a game for defense, or poor offense, if you prefer.
Until Barber's punt return the only scoring was field goals of 27 yards by Daluiso and 41 yards by Richie Cunningham. Cunningham, however, also missed from 48 and 41 yards, the latter on the final play of the third quarter, when the Giants called time out to force him to punt into the tricky Meadowlands winds of 10-to-20 mph.
Smith, who needed 25 yards to get to 10,000, got there on his touchdown run. But he finished with just 26 yards on 22 carries as the Giants totally shut down the Dallas running game.
But Troy Aikman came through late for Dallas, going 20-of-33 for 266 yards.
Graham, who had 111 yards until the final play, finished 15-of-21 for 183 yards in his first game back after missing last week with a concussion.
Still, it was mostly defense.
The Cowboys sacked Graham five times and the Giants got Aikman three times as both quarterbacks were constantly under pressure.
Dallas also had a number of injuries, the most serious to rookie wide receiver Wane McGarity, who separated a shoulder tackling New York's Sam Garnes after Garnes intercepted a pass at the New York 7 to thwart a Dallas drive in the second quarter.
That led to an 80-yard drive that led to Daluiso's first field goal. The Giants eschewed a fourth-and-1 at the Dallas 10 to try the kick.
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