Romo's 49-yard touchdown pass to Miles Austin and Barber's 3-yard run helped Dallas build a 14-0 first-quarter lead, and the Saints never led.
Romo completed 22-of-34 passes for 312 yards without an interception.
Barber gained 62 yards on 17 carries.
Austin caught seven passes for 139 yards.
Felix Jones gained 58 yards on 14 carries, while Nick Folk booted a 44-yard, second-quarter field goal for the Cowboys (9-5), who had lost two straight.
Drew Brees led New Orleans, passing for 298 yards with one touchdown and one interception. He also fumbled twice, and the Cowboys recovered both.
David Thomas made eight receptions for 77 yards.
Marques Colston had five catches for 86 yards, and Garrett Hartley kicked a 34-yard, second-quarter field goal for the Saints (13-1), whose 336 yards were their lowest single-game total this season.