The MLS outfit opened their campaign in front of their home fans and sent them away happy as goals in either half from Saer Sene and Lloyd Sam proved too much for the visitors.
Sene took advantage of a Tim Cahill assist to open the scoring after just 11 minutes with a well-placed effort.
However, the hosts were unable to double their lead until late in the game.
Sam finished at the near post after 70 minutes to kill the game off and end FAS' hopes of progression from Group Three - they bow out after three defeats from as many games.
"I think we proved tonight that we're taking this [competition] seriously," Red Bulls coach Mike Petke said after the game, as his side now seek to reel in leaders and MLS rivals Montreal Impact.
Deportivo Saprissa had hat-trick hero Ariel Rodriguez to thank for their 3-0 victory over Real Esteli in Costa Rica.
The Costa Rica international had his first of the game after 18 minutes and added to his tally twice more in the final 16 minutes to move his side top of the table.
Nicaragua's Real Esteli sit second in the Group Two table.
Comunicaciones and America shared the spoils in a 1-1 draw at the Estadio Cementos Progreso in Guatemala.
Oribe Peralta opened the scoring after five minutes in the Group Eight clash as the Mexican side appeared on course for a second win of the campaign.
However, Rolando Blackburn levelled shortly before the half-hour mark as the sides remain level on four points in Group Eight.
