A penalty from Cucho Hernandez and goal from Yaw Yeboah gave the Crew a 2-0 lead at the interval before Denis Bouanga pulled a goal back for LAFC in the 74th minute.
In what may have been the final game before retirement for former Italy captain Giorgio Chiellini and possibly a last appearance for LAFC for Mexican star Carlos Vela, last year's champions were convincingly beaten.
Columbus, coached by little-known Frenchman Wilfried Nancy, dominated the game for long stretches while Steve Cherundolo's LAFC were unable to unleash their trademark counter-attacks.
After a frantic start, in front of a loud, capacity 20,802 home crowd, Columbus quickly settled into their possession game thanks to the calming influence of veteran midfielder Darlington Nagbe.