The Falcons took a five-game lead in the NFC South with five games to play.
Atlanta had a first-and-goal at the New Orleans six-yard line when Warrick Dunn fumbled the ball away with 3:03 remaining.
But the Falcons held, regained the ball at the Saints' 47 and Vick needed just two plays to get his team in the end zone. He first hit Crumpler with a 27-yarder and then found him for the winning touchdown.
Vick completed 16 of 29 passes for 212 yards and two interceptions. He also scored a rushing touchdown and ran 69 yards on 10 carries.
Atlanta has won four straight, improving to 9-2.
New Orleans fell to 4-7, the same record owned by the other two NFC South members -- Tampa Bay and Carolina.