Within 10 minutes New York led by two goals, with Dru Yearwood finding the back of the net first when his shot from outside the box caught a deflection to wrong-foot the goalkeeper, before Aaron Long headed home Lewis Morgan's indirect free kick.
A costly mistake deep in defence gifted a goal back to the Rapids in the 21st minute, with Diego Rubio curling his right-footed effort into the far post, but the Red Bulls seemed to have all the answers, with Morgan scoring from the penalty spot seven minutes later to lead 3-1 after a half-hour.
Colorado ensured they would go into the break only trailing by one after a well-worked move from a throw-in resulted in a powerful low drive from outside the box by Keegan Roseberry in the 38th minute.
There were over 30 minutes of scoreless action to start the second half, before the Rapids seized control of the contest with two quickfire goals. Collen Warner equalised in the 77th minute, before substitute Michael Barrios put the visiting side in front three minutes later off a Rubio assist.
Dantouma Toure completed the comeback by making it 5-3 in the 89th minute, with a stoppage-time penalty to Tom Barlow providing the game's ninth goal.
Meanwhile, it was a very different contest at Lumen Field where the Seattle Sounders defeated Dallas FC 1-0 at home.
The only goal of the contest came from a first-half penalty, with captain Nicolas Lodeira calmly slotting it past the Dallas keeper in the 39th minute.
Cristian Roldan was at the centre of everything for the Sounders. Only two players completed more than 50 passes, and Roldan separated himself from the rest with 76 completed passes at an accuracy of 93 per cent.
With the win, the Sounders jumped up to sixth in the Western Conference, still three points behind fourth-placed Dallas, but with a game in-hand.