Goals from Yacine Brahimi, Hector Herrera, Casemiro and Vincent Aboubakar, combined for a 4-0 win at the Estadio do Dragao, and continued Basel's poor form in Portugal as their coach Paulo Sousa endured a miserable return to his homeland.
The first leg had finished 1-1 three weeks ago in Switzerland.
Porto coach Julen Lopetegui said he was both happy and proud at his side's passage into the last eight.
"You have to applaud the entire team for this fantastic performance against a very difficult opponent who came here determined to qualify," said Lopetegui.
"We deserve to be in the Champions League quarterfinals even if we came through qualifying. I am very happy and very proud of my players who showed a lot of character." he added.
The Swiss champions have yet to win on six trips to Portugal while Sousa, a former Benfica and Sporting Lisbon midfielder, saw his side fail again in their bid to reach the quarterfinals for the first time.
Algeria's Brahimi eased the nerves of the home fans and put the two-time champions in charge with a precision free-kick after just 14 minutes.
Former Argentina international Walter Samuel, who won the 2010 Champions League with Inter Milan, committed the offence that led to the setpiece and Brahimi made no mistake from just outside the box.
The 25-year-old, curled the ball past the wall as Basle 'keeper Thomas Vaclik stood rooted to his line and was well beaten to the left of his goal.
Samuel was later sent off in the final minute for a second yellow card.
Porto's Brazilian captain Danilo was then carried off with a worrying injury, and left the stadium in an ambulance, when he collided just outside the penalty area with his own goalkeeper and compatriot Fabiano.
Hector Herrera then scored his third goal of the competition with a neat right-foot shot from the edge of the area two minutes after halftime as Basle continued to crumble.
Brahimi again played a key role as his surging run and lay-off to Herrera laid the foundations for the goal.
"We showed a lot of confidence right from the start of the match and we were determined and conscious of what we had to do," said Herrera.
"I think that the result shows that the team played very well but that's the Champions League, you can't be distracted because any team can hurt you." he added.
As Basle struggled to stay in touch, the evening went from bad to worse for the visitors on 56 minutes with another precision strike from Casemiro.
The Brazilian hit a long range free-kick that flew past Vaclik and killed off any chances of a Basle comeback.
Cameroonian Aboubakar put the gloss on a fine night's work for the Portuguese champions with their fourth goal on 76 minutes with another high quality strike from 25 yards.
Porto, who won the competition in 1987 and 2004, now go into the hat for the quarterfinals for the first time since they were beaten 3-2 in the last eight by Manchester United in 2009.