LAFC leads the Western Conference with an 8-1-3 record. Columbus (5-7-1) lost for the sixth time in seven matches.
Kaye gave LA the 1-0 lead in the 37th minute before Diomande scored off a defensive blunder in the 88th minute and Vela scored his MLS-best 12th goal in the 90th.
Los Angeles, the stop scoring team in the league, was held to a scoreless draw vs. Chicago in its last match and couldn't breakthrough against the Crew until a fortunate play by Kaye for his third goal.
He took an entry in the box from Latif Blessing and appeared to try a pass back to him. Kaye instead mishit the attempt and caught goalkeeper Joe Bendik leaning forward as the ball rolled past him to the left post.
Bendik made amends two minutes later with a point-blank save of Diego Rossi.
The Crew was without U.S. goalkeeper Zack Steffen, who sustained a foot injury during the 3-1 win vs. the Los Angeles Galaxy on Wednesday. Bendik's previous appearance this season was a 3-0 loss at Philadelphia on March 23.
Diomande extended the lead when Jonathan Mensah's back pass to Bendik went awry. Josh Perez sent a shot off the crossbar and Diomande cleaned up the rebound. Vela then followed with a clinical finish two minutes later.
LA has allowed eight goals this season and is 6-0-0 when scoring first and leading at halftime.
The Crew starts a two-game trip with a match at Minnesota United in May 18.
LAFC will host FC Dallas at home Thursday.