Disclaimer: I know perfectly well that the 2005 Cowboys, 2012 Cowboys, and 2014 Cowboys had other losses to blame that could have kept them out of the playoffs, too.
In 2005, the Cowboys were leading the Commanders 13-0 at home on Monday night but Washington then shockingly came back to stun Dallas, 14-13. The Cowboys eventually finished 9-7 that year and missed the playoffs. All other things being equal, if they had beaten Washington, they would have been 10-6 and the Commanders would have been the team missing out at 9-7.
Side note: There were two huge, easily-preventable defensive lapses that cost the Cowboys dearly. First, Dallas allowed Mark Brunell to scramble for 23 yards on third-and-26; the Skins then converted the short fourth-and-3. Later on, Dallas allowed Washington to convert a 4th-and-15 late in the fourth quarter. Also, because of this defeat, the Commanders won their only playoff win of the last fourteen years. If they had lost to Dallas, they'd now have a playoff drought exceeding Dallas' from 1996-2009.
In 2012, the Cowboys, playing at home on Thanksgiving, allowed the RG3-led Commanders to jump out to a dominating 28-6 lead. The Cowboys fought back valiantly in the second half but couldn't overcome the deficit, falling 38-31 to Washington. The Cowboys were later on defeated in the season finale again by Washington and thus missed the playoffs. All other things being equal, if they had beaten Washington at home on Thanksgiving, they would have been tied 9-7 with the Commanders but awarded the NFC East division title due to the common-opponent tiebreaker.
In 2014, the heavily favored Cowboys were hosting the Commanders but shockingly lost, 20-17. Unlike 2005 and 2012, this loss is unlikely to mean that the Commanders will overtake the Cowboys in the standings; Washington is simply too bad this year. But the loss to the Commanders was immensely damaging in two ways; first of all, it was a loss, and a loss to an NFC East foe; second of all, Romo was injured, which led to Weeden starting against Arizona, and Weeden's horrible performance lost the game for Dallas. So this Washington loss really caused two losses instead of just one.
Instead of potentially being 8-2 or 9-1 entering the bye week and being a legitimate contender for the conference's top seed, the Cowboys are now fighting for their playoff lives to even get a wild-card berth, especially since the Cards and Niners both hold the tiebreaker over Dallas.
If the Cowboys miss the playoffs this year, the Washington defeat WILL almost certainly be our main cause of regret.