Seattle was clinging to a four-point lead when Browner scored, wrapping up the road victory for Seattle (2-3).
Tarvaris Jackson posted a 15-of-22 passing efficiency, compiling 166 yards with a touchdown and an interception.
Charlie Whitehurst took over behind center after Jackson left in the third quarter with a bruised right shoulder.
He went 11-of-19 passes for 149 yards and found Doug Baldwin on a go-ahead, 27-yard TD pass with 2:37 remaining. Baldwin finished with eight catches for 136 yards.
New York's Eli Manning tossed for 420 yards but was picked off three times.
Victor Cruz hauled down eight passes for a career-high 161 yards and a score for the Giants (3-2).