RSL knocked LAFC out of the MLS playoffs in 2018, prompting LAFC's Latif Blessing to promise that it was "payback time" earlier in the week.
But Damir Kreilach scored from the spot to make it 1-0 for RSL 35 minutes in, denting those hopes.
LAFC did respond five minutes later, Diego Rossi netting to pull his side level, but Bob Bradley's side left it late for a winner.
Walker Zimmerman was the man for the moment, netting a stunner a minute into stoppage time to give the home side a dramatic victory.
It was the first league defeat of 2019 for the Red Bulls, who lost just twice at home in all of 2018 as they stormed to the Supporters' Shield.
Former Red Bull Sacha Kljestan was the hero on the night for the visiting side, knocking home a Joao Moutinho cross to give Orlando a 1-0 lead in the 73rd.
Carlos Ascues also came up huge for Orlando with an acrobatic clearance off the line when Daniel Royer looked to have leveled matters for New York with a header three minutes later.
The loss was the first at home for New York since July 28 of last year.
Michael Barrios opened the scoring for Dallas in the 35th, but Tommy Smith managed to pull the Rapids level in the 69th, setting the stage for a late winner for the home side.
Ryan Hollingshead provided that winner in the 82nd, pushing home a rebound from a Zdenek Ondrasek rebound to give Dallas its second win of the young season.
Accam grabbed both of his goals before the break, Ilsinho adding a third just after halftime to make sure of the result and hand the Union their first win of the season.