The swift exit for Miami means the loss was the final game for the club's veteran Argentine striker Gonzalo Higuain, the former Real Madrid and Juventus forward, who had announced he would retire at the end of this season.
NYCFC, owned by Manchester Cityâs City Football Group, will now face Montreal in the next round on Sunday.
That match had been moved from Cityâs normal home, Yankee Stadium, due to the primary residents, baseballâs New York Yankees, being involved in playoff competition of their own and the visitors started brightly.
Finnish winger Robert Taylor went close for Miami with a drive from a tight angle on the right which City keeper Sean Johnson did well to push over the bar.
But Miami looked far from comfortable at the back and their hesitance from a corner allowed Thiago Martins a free header which he put against the bar.
Higuain, who ended the regular season in fine scoring form, was inevitably the danger man for Phil Nevilleâs Miami and he had an effort disallowed for offside in the 23rd minute after slotting home a lovely through ball from Christopher McVey.
New York then saw Miami defenders twice hit their own woodwork in the same bizarre move when a chipped effort from Brazilian Heber was cleared against the bar by Aime Mabika and McVey's misdirected clearance struck the post.
Miami had been sloppy with possession and were fortunate to go in goalless at the interval but City's superiority paid off in the second half.
Higuain went close with a drive from the left which had Johnson at full stretch but that was the last real threat from the visitors.
Gabriel Pereira opened the scoring with a fine curling finish after being set up by the bustling Santiago Rodriguez who held off two Inter defenders before finding the unmarked Brazilian.
Maximilinao Moralez then doubled the lead, slotting home after a long passing move ended with a clever back-heeled pass from Anton Timmerholm.
The win was secured when Heber added the third in stoppage time, finding the target at the second attempt after a poor back pass from Jean Mota.