Atlanta became the fourth team in playoff history to run for more than 300 yards. The Falcons will either host Minneapolis or visit Philadelphia in next week's NFC Championship contest.
The Falcons emerged from a wild first half with a 28-14 lead and then dominated the mistake-prone Rams during the final two quarters.
With St. Louis threatening to cut its deficit to 14 points, quarterback Marc Bulger fumbled the ball away on the first play of the fourth quarter to end any doubt.
Vick threw touchdown passes of 18 yards to Alge Crumpler in the first period and six yards to Peerless Price in the third. Dunn scored on first-half runs of 62 and 19 yards.
Allen Rossum contributed a key 68-yard punt return for a touchdown late in the first half.
Dunn gained 142 yards on 17 carries and Vick had 119 yards on eight.
Atlanta had 327 total rushing yards. Chicago holds the playoff record with 382, a mark set 65 years ago.