Walter Benitez's men needed something special to win by four goals and depose Martinique as one of the top-two best third-placed sides in the group stages but produced the goods in dramatic fashion at Rentschler Field in Connecticut on Tuesday.
An Ariel Martinez hat-trick set up a grandstand finish and, although they were denied by the offside flag in the first minute of injury time, the Cubans advanced to the final eight when Yenier Molina tucked away the decisive goal 60 seconds later.
It was Cuba's first win in 14 matches at the continental competition.
Earlier, Martinez dominated the Belize defence with his lethal left boot.
The 27-year-old winger's first goal in the 38th minute was a low strike into the bottom left corner of Belize's net.
Despite entering the half-time interval well short of their required four-goal margin, there was renewed hope in the 62nd minute when Martinez fired high past Belize's goalkeeper Woodrow West from close range after dancing past three defenders.
But still the odds were against Cuba as they entered the final 10 minutes needing two goals to earn progression to the final eight.
They benefitted from the dismissal of Belize defender Ian Gaynair in the 81st minute and Martinez made the Central Americans pay shortly after.
Martinez completed his treble with a right-footed shot from close range after a poor turnover in Belize's defence.
Cuba then thought they had the goal they so desperately desired in the closing stages but it was incorrectly ruled out for offside.
However, with less than a minute to play, Martinez crossed to the back post to find Liban Perez, whose header back across the face found Molina, who tucked home the goal to spark wild celebrations.
It ensured Cuba finished in third place in Group C on three points - with an identical record with Group A's third placegetter Martinique, although the Cubans scored more goals in their three group games to clinch a quarter-finals berth.
In the other match at Rentschler Field, the United States sealed top spot in Group C with a 1-0 win over Costa Rica.
With both sides already qualified for the quarter-finals, the result was academic but the tournament hosts became the only team to enter the knockout stages with three wins from three after Brek Shea's 82nd-minute goal, which was set up by a sublime pass from Landon Donovan.
And USA head coach Juergen Klinsmann priased his side's attitude after the match.
"I think we saw an entertaining game, a game that we pushed towards the end in order to win it and the team really deserves a huge compliment because we could have taken it easy, saying: you know what, we're first anyway in the group and a tie is fine," said the former Germany international
"But a tie is not fine with us. So the team pushed it, they ground it out until the last minute and deserved another goal."