The Cubans needed to score four goals to surpass Martinique as the second ranked third place team.
They did so in the final moments of the match, three minutes into added time, clinching their second quarterfinals berth.
Liban Perez headed a cross from the right wing to Marquez Molina, who gently guided a right-footed shot into the net.
Cuba took a 1-0 lead in the 38th minute after Martinez made a diagonal run along the 18-yard box before hammering a shot from 22 metres inside the far post.
The 27-year-old forward added his second in the 63rd minute, finishing a long run with a blast from eight metres to give Cuba a 2-0 lead.
Martinez netted his third in the 84th minute, taking a pass from Alberto Gomez and slotting it past Belizean goalkeeper Woodrow West.
Cuba will meet Panama on Saturday at the Georgia Dome in Atlanta.
Meanwhile five minutes after coming on a substitute, Brek Shea scored his first career goal to lead the United States to a 1-0 victory over Costa Rica also at Rentschler Field on Tuesday.
The U.S., which extended its winning streak in all competitions to a record eight games, captured the group title and will face El Salvador in the opener of Sunday’s quarterfinal doubleheader at M&T Bank Stadium in Baltimore.
