The previously winless Rams made the debut of interim head coach, Jim Haslett, successful following a bye last week. On Sept. 29, Haslett replaced the fired Scott Linnehan.
Brown also booted field goals of 51, 25 and 44 yards, plus an extra point.
Oshiomogho Atogwe scored on a 75-yard fumble return with nine seconds left in the first half, and quarterback Marc Bulger completed 15-of-26 passes for 136 yards, including a key 43-yarder to running back Donnie Avery during the game-winning drive.
Washington running back Clinton Portis scored twice. He rushed 21 times for 129 yards.
Wide receiver Antwaan Randle-El caught five passes for 87 yards. Quarterback Jason Campbell went 18-for-26 for 208 yards, but lost a critical second-quarter fumble.
The Redskins, whose four-game winning streak ended, lost four fumbles in all.