The Falcons, who had only won two play-off games in their 33-year history, struck down the visitors in style with Jamal Anderson rushing for 113 yards and two touchdowns.
Dan Reeves, the Atlanta coach who was in charge on the sidelines for the first time since his heart surgery last month, said: "The whole year has been incredible.
"I can't say enough about what a good bunch of guys. They've really sacrificed and to get to this point is certainly beyond our wildest dreams but now we've got a chance. We just need to take it to the next step."
In Saturday's AFC Divisional play-off game the Denver Broncos, the reigning Super Bowl champions, overwhelmed the Miami Dolphins 38-3.
Terrell Davis, held to a season-low 29 yards by the Dolphins three weeks ago, ran for two touchdowns and 199 yards, the fourth-best rushing performance in NFL play-off history. It was the Dolphins' heaviest play- off loss and means they have not won a play-off away from Miami since 1972.
The Broncos now play the New York Jets - 34-24 winners over the Jacksonville Jaguars - in Denver next Sunday for a place in the Super Bowl.