Kansas City went up 1-0 in the 12th minute when Kei Kamara hurried down the right sideline and found Arnaud in the center of the box. Arnaud held off defender Omar Gonzalez and slipped the ball inside the goal.
Conrad added an insurance tally in the 70th minute, redirecting an Arnaud free kick off the right post and across the Los Angeles goal line.
The Wizards (7-9-5) continued a late-season playoff push, picking up their 14th point in seven games and climbing to within three points of a playoff spot.
The Galaxy (13-5-4) allowed Columbus and Real Salt Lake to climb closer for the best record in the MLS. Los Angeles had 43 points, Columbus 41 and Real Salt Lake 40 following Saturday's games.