Benfica overturned a 3-2 deficit from the first leg in Russia to progress 4-3 on aggregate after dominating possession and restricting Zenit to sporadic counter-attacks.
Benfica kept pouring forward and the tactic paid off when the energetic Pereira scored from close range in first-half stoppage time after a neat backheel from Belgium international Axel Witsel.
Substitute Nelson Oliveira, who made his Portugal debut against Poland last week, grabbed the second goal just before the final whistle.
Benfica reached the Champions League quarterfinals for the first time since 2006 when they knocked out Premier League Liverpool in the round of 16 before losing to Barcelona in the quarterfinals.
Benfica dominated the first half and got their reward when Witsel's first shot was parried but he kept his cool to lay the rebound off for Pereira to slot in.
Man-of-the-match Witsel moved further forward due to the absence of first-choice playmaker Pablo Aimar and he rose to the challenge, not only winning many balls in the midfield but also acting as the prime creator for Benfica's attacks.
Zenit employed a more physical approach and seemed happy to concede most of the possession, opting instead for hitting Benfica on the counter-attack.
Benfica finished off the visitors in second-half stoppage time when the promising Oliveira scored with a clinical strike.