Kitna completed 13-of-22 passes and Dez Bryant was stellar, making three catches for 104 yards as the Cowboys broke a five-game losing streak and gave Jason Garrett a victory in his first game as the Dallas coach.
Felix Jones contributed a 71-yard touchdown grab and Miles Austin also caught a scoring pass for Dallas (2-7).
Eli Manning responded with 373 passing yards and two touchdowns, going 33-of-48 with two interceptions for the Giants (6-3), who saw their NFC East lead shrink to a half-game over Philadelphia.
New York got 73 rushing yards from Ahmad Bradshaw. Mario Manningham and Kevin Boss each made a TD reception in the loss.
New Meadowlands Stadium was briefly plunged into near-total darkness due to a lighting malfunction in the third quarter. Officials did not immediately give an explanation for the incident.