Beisel picked up the ball and scored after Sean Morey blocked a punt by Mat McBriar.
The Cowboys' Nick Folk had sent the game into overtime with a 52-yard field goal.
Dallas was forced to punt from its own 15 in the extra session, setting up Arizona's block and winning touchdown.
Kurt Warner passed for 236 yards and two touchdowns for Arizona (4-2).
Steve Breaston caught eight passes for 102 yards and a touchdown and Larry Fitzgerald caught five for 79 yards and another Cardinals' score.
Dallas quarterback Tony Romo completed 24-of-39 passes for 321 yards and three touchdowns.
Marion Barber caught a 70-yard touchdown pass, totaled 128 yards receiving and added 45 yards rushing, and Patrick Crayton pulled in a 55-yard scoring pass for the Cowboys (4-2).