Goncalo Ramos, brought into the team for the dropped Cristiano Ronaldo, scored a brilliant hat-trick to more than justify Portugal manager Fernando Santos's controversial decision.
Ronaldo had angered his coach with his reaction to being substituted in the group game against South Korea.
But 21-year-old Benfica forward Ramos proved he is ready to take on the mantle of Portugal's talisman as he lashed a superb finish from an almost impossible angle past Swiss keeper Yann Sommer to get the ball rolling after 17 minutes.
At the other end of the age scale, 39-year-old Pepe headed Portugal into a 2-0 lead after 33 minutes. Ramos then bagged his second early in the second-half before defender Raphael Guerreiro made it four.
Manuel Akanji got one back for the Swiss just before the hour before Ramos completed his hat-trick with a lovely, chipped finish. Substitute Rafael Leao made it 6-1 with a thunderous shot in added time.
Portugal will now face Morocco in the quarter-finals on Saturday after the North Africans beat Spain 3-1 on penalties.
Walid Regragui's team are the fourth African side to make the World Cup last eight and the first from the north of the continent to do so. They are the first Arab team in history to reach the quarter-finals.