Gibson received about 40% of the vote to beat out Carolina Panthers outside linebacker Jeremy Chinn, who got 36% for returning two fumbles for touchdowns.
After the regular season, five players will be nominated for Rookie of the Year honors. Fans will be able to vote for the winner on https://www.nfl.com/voting/rookies/ throughout January.
Gibson, who split time as a rusher and a receiver at Memphis, seems to be getting more and more comfortable as a traditional NFL running back. In his previous four games entering Thursday, Gibson rushed for 287 yards (5.2 per carry) and found the end zone five times.
He was even better against the Cowboys, compiling 115 yards on 20 carries and adding 21 yards on five receptions. He also scored three touchdowns -- the first coming at the end of the opening quarter and his final two breaking open Washington's eventual 41-16 triumph.
"The thing about him is he's got so much potential and ability," head coach Ron Rivera said of Gibson after the game. "He's a lot of fun to watch. I'm pretty excited about who he's developing into."
For the season, Gibson ranks second behind wide receiver Terry McLaurin with 878 scrimmage yards (645 rushing and 233 receiving). He has also scored 11 rushing touchdowns, which is the third-most in the NFL and two shy of the franchise's rookie record that Alfred Morris set in 2012. Gibson has five more games to try and break that mark.
"He's very special, and I don't think he knows how good he can be," McLaurin said. "He's a very hard worker -- he doesn't really say much -- and just to see the way he's grown and he's matured over the season. I feel like he has a really good feel for where the holes are, the O-line is giving him plenty of time to make those cuts, and when he hits them, you see that 4.3 speed, so we need that explosiveness out of our backfield."