The result takes the Zoros two points clear of sixth-placed New England Revolution, although they have played one game more than the Massachusetts outfit.
However, the rebound fell to Casey, who showed no mercy in firing into the net and wheeling away amid a chorus of boos.
Nielsen denied Farfan for a second time in the 63rd minute, this time with an outstretched foot, after the midfielder got himself on the end of a Sheanon Williams cross.
The hosts dominated possession and had plenty of efforts on goal, but they were unable to break down a resolute Union side, who defended in numbers to eke out three points on the road.
The Wizards could have gone level on points with leaders New York Red Bulls had they won, but instead they remain in second place - three points ahead of Montreal Impact.