The quarterfinal match -- a Hudson River derby between the top two teams in Major League Soccer's Eastern Conference -- ended 0-0 after the first-half despite end-to-end action with the hosts looking particularly dangerous in the opening 45 minutes.
NYCFC was left to play the remainder of the match with 10 men when Thiago Andrade was shown a straight red for violent conduct and the Red Bulls took advantage of the extra player, doubling their lead soon after with a tidy close-range finish Luquinhas.
Omir Fernandez scored the Red Bulls' third in second-half stoppage time to add further gloss to the scoreline and send Gerhard Struber's side into the semifinals.
The Red Bulls will now await the results of Thursday's draw for the semis and finals to determine their next opponent. The Sacramento Republic reached the semifinals by beating the LA Galaxy 2-1 on Tuesday, Sporting KC went through with a 6-0 Wednesday night thrashing of third-tier Union Omaha and the last quarterfinal match between Orlando City SC and Nashville SC scheduled for June 29.