Joe Flacco found Brown with the clutch scoring strike after the lead had changed hands five times in the final quarter of a game marked by heavy snowfalls and frigid temperatures.
After Minnesota had grabbed a 12-7 early in the fourth, Dennis Pitta caught a 1-yard touchdown pass with 2:07 left and Flacco completed a 2-point conversion for the Ravens (7-6).
The Vikings responded almost immediately on Toby Gerhart's 41-yard scoring run for a 19-15 lead with 1:27 on the clock.
Baltimore's Jacoby Jones then returned the ensuing kickoff 77 yards for a score and a Baltimore lead with 1:16 to play.
But Minnesota (3-9-1) came right back for a 26-22 lead on Cordarrelle Patterson's 79-yard catch-and-run with 45 seconds left.
On their final drive, the Ravens took advantage of pass interference penalty on the Vikings' Chad Greenway and Brown finally ended it with a difficult scoring grab at the back of the end zone.
Minnesota's Adrian Peterson left the game in the second quarter with an ankle injury and did not return.