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Dallas looked to be in control in the first quarter as Randles powered over from one-yard out, but Devonta Freeman's score reduced the deficit to seven at the end of first quarter.
Darren McFadden went over for the hosts' third touchdown of the day early in the second quarter before Freeman returned the favour as the Falcons trailed 21-14.
Randle then scored his third TD of the game as the Cowboys looked to be heading for a healthy half-time lead before Matt Bryant kicked a field goal for Atlanta.
The Falcons' hopes of a comeback in Arlington were given a timely boost when Matt Ryan found Julio Jones, who grasped a 45-yard touchdown pass before the visitors managed to make a successful two-point conversion.
Atlanta continued to threaten on offense as Freeman picked his third score of the game to give the away side the lead.
Dan Quinn's men then sealed victory in the dying minutes as Jones completed a good day with 164 yards from 12 catches and his second touchdown.