The first touchdown of the match went the way of the Bears as Matt Forte strolled into the end zone from two yards out.
The Redskins took the lead with two scores in quick succession at the start of the second quarter. First Roy Helu found a way through the defense for a 14-yard score before Brian Orakpo collected a spilled Jay Cutler pass and returned it 29 yards for a touchdown.
Devin Hester levelled the scores again later in the quarter, collecting a punt and returning it 81 yards for a touchdown thanks to some good blocking from his teammates.
Washington would take a lead into the halfway stage, however, as Robert Griffin III floated a pass out to Jordan Reed in the end zone, and the rookie tight end got his feet down just in time to complete the score.
Forte restored parity for the Bears once more midway through the third quarter as he went straight down the middle of the field before side-stepping a defender and running in for a 50-yard touchdown.
The pendulum swung back in favour of the Redskins with 3:26 left of the quarter, however, as Helu dodged a number of tackles to go in for his second touchdown of the match.
Forte went one better by claiming his third score of the match at the start of the fourth quarter, bringing the sides level for sixth time in the match with a six-yard run through the middle.
Washington regained the lead shortly afterwards, however, as Griffin found Aldrick Robinson at the front of the endzone with a fine 45-yard pass.
Chicago moved back within four points courtesy of a 49-yard Gould kick before taking the lead with 4:02 left on the clock when Josh McCown - in for the injured Cutler - found Martellus Bennett in the end zone with a seven-yard pass.
However, the Redskins took the lead for the sixth and final time with just 49 seconds left of the match as Helu ran in from three yards for his third touchdown of the day to secure a dramatic win for his side.