The Cowboys entered the game on a six-match winning streak but were forced to play catch-up throughout the contest at their AT&T Stadium in Texas, with Washington eventually claiming victory in overtime.
Washington led 3-0 at the end of the opening quarter thanks to another Forbath kick but trailed 7-3 at half-time.
They then took the lead two more times before Dallas forced overtime, with Forbath slotting a 40-yard kick to secure the win on the road.
Dallas' starting quarterback Tony Romo was forced off the field in the third quarter with possible concussion but returned in the fourth term and completed the Cowboys' overtime drive, although he was unable to turn the game in his team's favour.
It was just the Redskins' third win of the season with Colt McCoy leading the visitors to victory in his starting debut for Washington, with the 28-year-old quarterback completing 25 of 30 passes for 299 yards and rushing for a touchdown in the fourth term.
McCoy joined Washington from the San Francisco 49ers in April and only made his debut last week when he came off the bench to help the Redskins to a 19-17 win over the Tennessee Titans.
Romo dragged Dallas back into the contest in the second quarter, leading the Cowboys on an 80-yard drive until he picked out Dez Bryant, who powered his way to the end zone.
The Redskins claimed a 10-7 lead in the sixth minute of the third quarter thanks to Alfred Morris' rushing TD and Forbath's conversion, before a field goal from Dan Bailey dragged Dallas level.
McCoy's touchdown early in the fourth stanza helped the visitors to a 17-10 advantage but Dallas' second-string QB Brandon Weeden led the hosts on an 80-yard drive that ended in a pass to Jason Witten, making the score 17-17.
After Forbath's overtime field goal, the Cowboys failed to earn a first down, moving the ball just seven yards, as Washington celebrated victory.
The result left Dallas with a 6-2 win-loss record at the top of the National Football Conference's East division, while Washington (3-5) remain last.