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 following a neat backheel from Belgium international Axel Witsel.
"Benfica and its fans deserve to be in the quarter-finals, specially those around their twenties who never saw Benfica in its widest European glory," Benfica coach Jorge Jesus told reporters.
"We knew that Zenit was not going to mind that we kept most of the ball, therefore we needed to score first so they would try and come out at us," Jesus added.
Substitute Nelson Oliveira, who made his Portugal debut against Poland last week, grabbed the second goal just before the final whistle.
Benfica, European champions in 1961 and 1962, repeated their feat of six years ago when they knocked out Premier League Liverpool in the round of 16 before losing to Barcelona in the quarter-finals.
Benfica dominated the first half and got their reward when Witsel's close-range shot was parried but he kept his cool to lay the rebound off for Pereira to slot in.
"Getting that first goal at the end of the first half brought us tranquility, the team deserved the win," Pereira said.
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.
Russian champions Zenit employed a more physical approach and seemed happy to concede most of the possession, opting instead for hitting Benfica on the counter-attack.
"We tried to defend our advantage the best we could but now we will just have to keep working to be Russian champions," said Zenit defender Bruno Alves.
Benfica finished off the visitors in second-half stoppage-time when the promising Oliveira scored with a clinical strike.
"The coach told me space would be opening up in the second half and that there would be chances for me to score and so it was. We put in a great display," said the 20-year-old Oliveira, dubbed "Portugal's Cantona" in reference to former Manchester United and France striker Eric Cantona.
