Walter Benitez's men only made it through to the last-eight meeting with Panama thanks to a stoppage-time goal, which meant they went through as the second-best third-placed side in the group stages.
Following their 4-0 win over Belize, which saw Ariel Martinez score a hat-trick before Jeniel Marquez grabbed the all-important goal, Benitez declared the victory "a great achievement" and "like winning the World Cup".
The quarter-final will be Cuba's first since 2003, when they were beaten 5-0 by the USA, and Benitez's men are likely to face a tough challenge when they meet group A winners Panama.
Julio Dely Valdes' side are unbeaten in the competition so far, winning two and drawing one of their games, conceding just one goal in the process.
After seeing off Mexico and Martinique, Panama drew a blank against Canada, knocking Antonio Fonseca's side out of the competition.
Dely Valdes said after the draw that he felt his team's fitness would be crucial to their overall chances.
"A rested team in these short tournaments is very important. It can be a determining factor on the field," he told the CONCACAF website.
He will be looking for joint-second-top scorer Gabriel Torres to continue his good form in the tournament so far.
Torres is level on goals with Cuba's Martinez and is the only member of his side to score at the competition.
The two countries have met 11 times in total, with Panama being victorious on six occasions and the sides' only previous Gold Cup meeting ending in a draw.
The winners of the last-eight tie will meet either Mexico or Trindad and Tobago in the semi-finals.