The Cowboys pulled ahead of the Cardinals 21-14 early in the fourth quarter on a touchdown pass from Dak Prescott.
Prescott hit Brice Butler on the 37-yard strike in the end zone.
The scoring play capped a five-play, 50-yard drive that took only 2:53.
Tie game
An over-the-should 15-yard catch by Cardinals receiver Larry Fitzgerald in the corner of the end zone tied it at 14 late in the third quarter.
Earlier in the drive, Fitzgerald also made a key 37-yard catch.
With that touchdown catch, Fitzgerald passes Marvin Harrison for eighth on the NFL's all-time receiving yards list (14,581).