The visitors needed just seven minutes to take the lead through Alvaro Saborio with Edgar Espindola adding strikes in the 12th and 22nd minutes to put Real Salt Lake in the driving seat.
Joel Lindpere pulled a goal back for the Red Bulls after 69 minutes but it was too little too late.
Real Salt Lake lie seven points behind Western Conference leaders the Los Angeles Galaxy and just one win away from a place in the play-offs.
A brace from Juan Pablo Angel saw Chivas come from behind to claim a 2-2 draw at DC United.
Dwayne De Rosario (39) and Ethan White (47) fired the home side in front but Angel's strikes in the 58th and 71st minutes rescued a point.
The Portland Timbers and the San Jose Earthquakes played out a 1-1 draw.
Portland were ahead after nine minutes through Kenny Cooper but Khari Stephenson equalised 20 minutes from time.