Romelu Lukaku's late first leg goal was ultimately the decisive strike in the tie as the two sides played out a goalless draw at the Estadio do Dragao.
Despite a simply astonishing and breathless finish to the game, the Italians heroically held on when Porto threw everything at them in injury time.
It means that Simone Inzaghi has led the Nerazzurri into this stage of the Champions League for the first time since 2011.
The omens looked good for Porto coming into the game, as they had won their last four home games against Italian sides in Europe, as well as never losing to Inter here in the Champions League.
However, in this competition the Dragons had lost six of the seven ties where they lost the first leg, while Inter have progressed on four out of five occasions after winning the first leg.
Porto were left reeling pre-game when it was announced Pepe would not be fit enough to start in defence.
Otavio was also unavailable for the hosts, meaning Canadian Stephen Eustaquio was forced to play in a less familiar role on the right of midfield.
Inter's only issues came at left-back, with Robin Gosens and Dalbert their only absentees, but the impressive Federico Dimarco started and showed his defensive capabilities all night on top of his renowned sensational crossing ability.
After losing at Spezia on Friday, Inter needed to sharpen up, especially defensively, and even though they held their own stoutly all night, Porto had the best of early exchanges.
Matheus Uribe's venomous strike flashed narrowly wide inside three minutes, before Eustaquio forced a smart save out of Andre Onana down to his right-hand post with a strike from the edge of the area.
Moments later, Inter had their first real venture forward, as Nicolo Barella was presented with a three-on-two opportunity and picked out Edin Dzeko, but he could only fire straight at Diogo Costa.
The Bosnian failed to affect the game at all from that point before being substituted, meaning he still has no goal in 10 games.
Back down the other end, Mehdi Taremi had a great opening on the left flank, but after cutting inside and aiming to curl it inside the far post, the Iranian got his effort all wrong and skewed it well wide.
There were shades of AC Milan's performance at Tottenham Hotspur last week in the Inter setup, and Inzaghi's side emulated their city rivals perfectly with multiple last ditch blocks while still looking relatively useful on the counter.
Porto remained dominant in the opening stages of the second half, apart from a Barella chance which flashed wide, but they struggled to fashion out any real chances of note until the latter stages of the game.
The more the second period went on, the further Inter dropped deep, and a substitute which saw Danilo D'Ambrosio replace Dimarco showed Inzaghi wanted a more defensive option in his back five to see the game out.
Half chances continued to come and go, as Marko Grujic tested Onana before Hakan Calhanoglu blazed over for Inter at the other end.
In a quite extraordinary passage of play during the seven minutes of additional time though, Inter were clinging on by their fingertips as Porto came forward with wave after wave of attacks.
Denzel Dumfries was in the right place at the right time to clear off the line in the fifth minute of injury time, and just seconds later Taremi struck the post after a vital touch from Onana in goal, with Grujic then hitting the bar on the rebound.
A late sending off for Porto's other Pepe was one final frustrating nail in the Portuguese side's coffin as Inter remarkably held on despite late carnage to draw 0-0 on the night.
Inter therefore join their city rivals Milan in the last eight and the two could meet when the draw is made on Friday, with the draw now left completely open from this stage onwards.
After reaching the quarter-finals in 2018-19 and 2020-21, it was one step too far again for Porto this season, and Sergio Conceicao looked furious at the final whistle, possibly scratching his head at how his side did not find the net in stoppage time.
Sitting eight points behind Benfica in the Primeira Liga, Porto face a tricky test away to Braga at the weekend to try to keep their improbable title challenge alive.
Inter meanwhile, remain second in Serie A despite their awful defeat domestically last time out, and they face Juventus in the Derby D'Italia at the weekend in one of the standout games across Europe coming up.