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City was in control throughout, creating 16 shots in the first half compared to just four for Miami, but it went into halftime tied at 0-0 after hitting the woodwork three times.
In one instance, a turnover from Miami goalkeeper Drake Callender handed City's Heber a golden opportunity to break the deadlock, but his chip was cleared off the line into the crossbar, with the rebound then being hacked off the post by a second defender.
City's dominance was rewarded in the second half, with Santiago Rodriguez brilliantly working his way into the Miami penalty box before dishing it off for Gabriel Pereira to finish in the 63rd minute, before Rodriguez also set up the second goal six minutes later with a clever backheel to Maximiliano Moralez.
Heber completed the rout in stoppage time to end the career of Miami's iconic striker Gonzalo Higuain, who was playing in his final season, with City now set to take on Montreal in the Eastern Conference semi-finals.
Later in the day, FC Dallas needed a penalty shoot-out to advance past Minnesota in the Western Conference.
Emanuel Reynoso's goal for Minnesota in the 53rd minute was canceled out by a 64th-minute strike from Facundo Quignon, and after 30 minutes of extra time, it all came down to penalties.
It was Minnesota's captain Wil Trapp who failed to come through with his side's second penalty, with his tame effort into the bottom-left corner being saved by Dallas' Maarten Paes, before Alan Velasco finished things in style with a cheeky panenka.