Kansas City (8-11-6) trailed 2-1 after 75 minutes, but Josh Wolff tied it in the 77th and Conrad's sprawling header three minutes later gave the Wizards their first home victory since a 3-1 win over New England on June 12. Dallas (7-12-6) has won just once in its last five games.
Santiago Hirsig scored his first MLS goal in the ninth minute, putting the Wizards up 1-0 with a header off Claudio Lopez's corner kick, before Dax McCarty and Jeff Cunningham each turned a mistake by Conrad into a goal.
McCarty scored in the 44th minute picking off Conrad's errant pass 25 yards from goal, making a short run and firing a low shot past Kevin Hartman from 20 yards.
In the 54th minute, David Ferreira stripped Conrad of the ball and drew Hartman to his right before slotting a pass to a wide-open Cunningham making a run up the middle.
Wolff, who has four goals in Kansas City's last three games, deflected second-half sub Zoltan Hercegfalvi's shot just inside the post to make it 2-2. Not long after that, Conrad took advantage of a mistake by Dallas' back line.
George John tried to head Lopez's corner kick away from the goal, but instead sent it looping toward Dallas keeper Dario Sala. Sala barely kept the ball from going in for an own goal, but it fell right in front of Conrad.