The eagerly-awaited contest went the visitors' way thanks to first-half goals from Fabian Espindola, Chris Rolfe and Perry Kitchen.
DC's triumph sees them climb to the Eastern Conference summit with 43 points from 24 matches, while Kansas drop to second with 42 points from 25 matches.
The Crew's victory was sealed by strikes from Justin Meram, Adam Bedell and substitute Aaron Schoenfeld.
The New York Red Bulls moved to fifth in the Eastern Conference courtesy of a 4-2 triumph against bottom side the Montreal Impact.
Braces from Thierry Henry and Bradley Wright-Phillips secured the three points for Mike Petke's side, while Dilly Duka and Andres Romero netted consolation strikes for the Canadian outfit.
In the Western Conference, first-half goals from Landon Donovan and Marcelo Sarvas ensured the LA Galaxy cruised to a 2-0 win against the Vancouver Whitecaps, who had Johnny Leveron sent off in the closing stages.