The ‘Skins (6-7) continue to hold a share of first place in the NFC East. Philadelphia defeated Buffalo, 23-20, on Sunday, to maintain their share of the lead.
New York (5-7) can jump back into the mix with a win Monday night at Miami.
Washington led 14-0 early in the second after touchdown runs by Alfred Morris and Kirk Cousins. An Alshon Jeffery 20-yard TD reception with 18 seconds left in the first half sent the game to the break with the Redskins up 14-7.
Cousins connected with Jordan Reed on a 5-yard TD pass with eight minutes left in the third to push the lead to 21-7.
But the Bears (5-8) would tie things up later in the third, getting a Jay Cutler-to-Zach Miller 9-yard touchdown pass and a Matt Forte 7-yard TD run.
Dustin Hopkins broke the tie with a 47-yard field goal at the 14:07 mark of the fourth.
Washington had two other long drives in the fourth quarter fizzle out around the Chicago 40. The second gave Chicago the ball at its own 20 with 2:19 left, and a 50-yard pass from Cutler to Jeffery after a false-start penalty set the Bears up on the edge of field-goal range.
A Forte 3-yard run and two incomplete Cutler passes forced the long field-goal try from Gould, which missed wide right.
Cousins was 24-for-31 passing for 300 yards and a touchdown with an interception. The effort, his fifth 300-yard game in 2015, pushed Cousins, in his first full season as a starter, past the 3,000-yard mark, to 3,006.
Jordan Reed had nine catches on nine targets for 120 yards for Washington.
Chicago outgained Washington, 377-374. Cutler was 19-for-31 passing for 315 yards and two touchdowns.