Trailing 2-0 from the first leg, Barcelona levelled the tie on aggregate thanks to two brilliant Lionel Messi strikes in the first half, before David Villa put them ahead in the tie early in the second period.[break]
Jordi Alba then made sure of the Catalans´ place in the quarter-finals for the sixth season running when he secured a 4-0 win on the night.
Meanwhile, Turkish giants Galatasaray beat Schalke 3-2 in Gelsenkirchen to advance 4-3 on aggregate and reach the quarter-finals for the first time since 2001.
No team had overturned a two-goal first-leg deficit in the Champions League without having the benefit of an away goal, so Barcelona´s task against Milan was an ominous one.
But the Catalans chose the perfect moment to produce their best performance in a long time, even if they had to survive a nervy ending.
They had the momentum from the moment Messi opened the scoring five minutes in, controlling and then instantly finishing brilliantly into the top corner from the edge of the area despite being surrounded by five Milan defenders.
The near-95,000 crowd then had their hearts in their mouths when M´Baye Niang broke through at the other end only to see his shot rebound to safety off the far post.
But, moments later, Messi smashed home his second through the legs of Philippe Mexes to level the tie on aggregate and take his overall tally in the Champions League to 58.
Villa clinically converted the third goal 10 minutes into the second half, but the home side endured some nervy moments towards the end before the indefatigable Alba broke through to secure the win in stoppage time.
It is the third time in eight seasons that Barcelona have knocked out Milan in the knockout stages of the competition.
"We believed from the first moment, from the moment the first leg finished," Villa told Spanish television TVE after the game.
"We were switched on right from the start. The early goal helped us, but we went looking for it straight from kick-off."
Milan coach Massimiliano Allegri admitted: "The pressure that Barcelona put us under in the first half-hour was devastating.
"We created chances but couldn´t finish them. Barca remain the best team in the world, with great players, players like Messi and Iniesta who can dribble past any defenders easily."
Schalke´s 1-1 draw in the first leg in Istanbul meant the Bundesliga side had a slight advantage going into the return in Germany and they took the lead on the night 17 minutes in when Roman Neustaedter netted from close range.
But the Turks turned the tie on its head before the interval, with Hamit Altintop striking from long range against his former club before Burak Yilmaz pounced on poor defending to net another five minutes later.
Remarkably, Yilmaz has now scored eight goals in eight Champions League games this season, making him the competition´s leading scorer alongside Cristiano Ronaldo.
Schalke now needed two without reply to go through, and they got one just after the hour when Michel Bastos netted after a Marco Hoeger shot had hit the bar, causing chaos in the visitors´ defence.
But the Galatasaray defence held firm, before substitute Umut Bulut made sure of their progress at the death, sparking wild celebrations among the huge visiting support.
"We played very well in the first half this evening," said Altintop.
"In the second half Schalke tried to put pressure on us and created a few important chances to score, but that is normal. Eventually, we got what we needed. I hope we get a good draw on Friday."
Barcelona and Galatasaray join Real Madrid, Borussia Dortmund, Juventus and Paris Saint-Germain in Friday´s last-eight draw.
The final two names in the hat will be decided on Wednesday, when Bayern Munich face Arsenal and Malaga host Porto.
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