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.
Celebrity Breakups and divorces of 2014 [PHOTOS]