Magee's goal came on his fourth and final shot of the match, a 14-yarder that beat United goalkeeper Joe Willis cleanly in the 85th minute.
Until then, a carom that first hit the chest of Fire midfielder Jeff Larentowicz, then hit DeLeon in the chest before bouncing off the turf and trickling across the goal line in the ninth minute was the game's only score. Willis had moved in the original path of the ball and couldn't reverse course in time.
Not counting the own goal, the Fire (3-7-2) have scored only eight goals in 12 matches this season, and DC (1-10-2) has only six in 13.
Fire goalkeeper Sean Johnson needed to make only one save for his third shutout of the season.
Chicago got its first MLS victory since beating Columbus 1-0 on April 20. The Fire were 0-3-1 in league play since.
The loss extended United's winless streak to 0-9-2, and kept them last in the Eastern Conference.
Magee, making his MLS debut for Chicago, was acquired from Los Angeles in exchange for the rights to Robbie Rogers last weekend. He was saved on a breakaway by Willis in the 83rd minute, two minutes before he scored his seventh goal of the season.