Trailing 1-0 from the first leg thanks to Galeno's wonder strike, the Gunners were not at their scintillating best in the first 45 but managed to level the tie on aggregate through Leandro Trossard's crisp finish.
No goals were to be had during a nail-biting second half, or in the additional 30 minutes, but David Raya was the Gunners' 12-yard hero, saving from Wendell and Galeno as Arsenal remained perfect from the spot, thus advancing to the last eight of Europe's elite tournament for the first time since the 2009-10 season.
As Porto kept faith with the same XI from the first leg, Arsenal boss Mikel Arteta brought Jorginho in for the injured Gabriel Martinelli, while Raya unsurprisingly demoted Aaron Ramsdale back down to the bench.
Inside the Emirates cauldron, the Gunners unsurprisingly began the contest on the front foot and tested Diogo Costa in the 13th minute; the Portuguese stopper saved Bukayo Saka's tame attempt before Martin Odegaard's low drive rippled the side netting.
However, Arteta's men notably lacked a sense of urgency going forward, and Porto surprisingly had more possession of the ball in the earliest exchanges, as well as having Arsenal exactly where they wanted them.
Top scorer Evanilson had a couple of sighters of goal, firing wide in the 16th minute before forcing Raya into a diving stop in the 23rd, and Sergio Conceicao's men were not afraid to press their hosts high in search of a greater aggregate advantage.
The gamesmanship and delays were also starting early from the Dragons, who were doing a brilliant job of agitating the home crowd, but a piece of playmaking brilliance led to Emirates lift-off in the 41st minute.
After winning a second ball high up the pitch, Odegaard combined with Trossard on the left-hand side, slipping his teammate in with an exquisite eye-of-the-needle pass before the Belgian did the rest, coolly picking out the far corner through Pepe's legs.
As well as bringing Arsenal level in the tie, Trossard's strike made him just the third Gunners player to score in each of their first three Champions League home games after Saka and Alexis Sanchez, and the hosts picked up where they left off at the start of the second half.
Clear-cut chances were still at a premium for the men in red, who forced a couple of corners to no avail and also conceded a 64th-minute free kick in a promising position, but Francisco Conceicao's effort was high, wide and not at all handsome.
In contrast, Odegaard had the ball in the back of the net with a delightful dink in the 67th minute, having benefitted from a piece of miscommunication from Pepe and Costa, but referee Clement Turpin had spotted a shirt pull from Kai Havertz on Pepe and correctly disallowed the goal.
Three moments later, a Porto three-on-three ended with Raya spilling a powerful drive from Conceicao at the feet of Evanilson, but Jakub Kiwior made a fantastic last-ditch block to deny the striker before the offside flag went up.
Not until the final 10 minutes would Arteta shuffle his pack - bringing on Gabriel Jesus for the leggy Jorginho - and the Brazilian almost made the difference with his first touch, but Costa made himself big at his near post and managed to deflect the ball behind.
The introduction of Jesus suddenly breathed a new lease of life into the Gunners, who also saw the hitherto quiet Saka test Costa with a curling drive in the 85th minute, and Odegaard agonisingly put the rebound wide from outside the box.
The Gunners captain then went down in the box under pressure from Pepe in added time, but Turpin saw nothing wrong with the incident, and Porto midfielder Alan Varela - a non-stop runner over the two legs - was stretchered off just before the final whistle signalled an extra 30 minutes.
The first half of extra time came and went with nothing to write home about, and barring one tame deflected Galeno strike - which fell in Raya's grateful grasp - it was an identical story in the second 15 as the dreaded penalties loomed.
After Odegaard, the attacking Pepe and Havertz all found the mark, left-back Wendell - who had neutralised Saka excellently all evening - saw his effort tipped onto the post by Raya, who subsequently breathed a sigh of relief when the ball bounced back off the post onto his legs but rolled away.
Saka, Marko Grujic - only just - and Declan Rice held their nerve too, but Porto's first-leg hero was thwarted by Arsenal's second-leg hero, as Raya palmed Galeno's effort away to spark Emirates ecstasy.
Snapping their streak of seven successive last-16 eliminations, Arsenal join Real Madrid, Manchester City, Barcelona, Paris Saint-Germain and Bayern Munich in the last eight, while two of Atletico Madrid, Inter Milan, Borussia Dortmund and PSV Eindhoven will complete the fascinating quarter-final selection on Wednesday night.