Perfectly poised after last month's 2-2 draw in Turin, Juve nicked an early lead on the night to go ahead on aggregate, with the ball put on a plate for Paul Pogba after an uncharacteristic mix-up in Bayern's defence - forced by the pressure of the Italians.
Before the half hour mark, a sloppy home side would see their two away goals cancelled out too as Juve went 4-2 up on aggregate. Alvaro Morata picked up the ball in the middle of the park after the Germans poorly gave it away, and his determined drive ended with him teeing up Juan Cuadrado to finish.
Robert Lewandowski gave Bayern a late glimmer of hope, heading home after Juve wasted a whole host of chances, and Thomas Müller followed this up with an almost identical goal in the first minute of stoppage time, taking the game to extra time in the most dramatic of fashion.
This gave the Germans the momentum they needed to limp over the finish line, with Thiago Alcántara on target off of the bench in the second half of added time.
Following that, just two minutes later, Juventus loanee Kingsley Coman smashed in a fourth for Bayern to put the cherry on the cake, his solo effort ensuring passage into the last eight.
Fearless Juve take the lead
Up against it after surrendering two away goals in Turin last week, Juventus got off to the perfect start at the daunting Allianz Arena, and it was a start they deserved after such fantastic early positivity.
The pressure they were applying to Bayern's back line in particular was more than encouraging for the fans and the team alike, and it was what caused a rare lapse in the hosts' defence.
Hesitation from David Alaba and misjudgement of the ball from Manuel Neuer - which all came from the pressure of marauding right-back Stephan Lichtsteiner - allowed the ball to be squared to Pogba, who calmly slotted into an empty net with only six minutes on the clock.
Juve weren't about to take the foot off the gas either. The best way to keep Bayern out is to push them back, not sit deep and invite pressure, and they did this as they surged forward for a second.
Italians get their tails up as they smell blood
The Bundesliga giants would eventually settle down into their typically dominant rhythm, but they were still lacking that cutting edge in those crucial moments as attempted passes continued to evade their targets and first touches bounced away unfavourably.
At the back, they were out-of-sync too, with that early goal having rattled them significantly. A poor clearance from Neuer struck Sami Khedira, and the former Real Madrid man fed Morata, but he was a yard offside as he coolly lifted the ball over the goalkeeper.
The linesman's flag had thwarted them this time, but Juve's pressure was continuing to be effective and increasing the likelihood of them getting another goal.
Franck Ribery, meanwhile, looked like Bayern's best bet of getting a goal back and restoring some order. The Frenchman whacked a volley wayward moments after Pogba's opener, and he came close with a cutely disguised header just before the half hour mark.
However, the team as a whole didn't deserve a goal in truth and it was Juve who earned the second of the night with a perfectly executed counter attack.
Sloppy in possession, as they had been for the entire of the evening's play up to this point, Bayern lost the ball in the middle of the park and were severely punished. Morata picked it up and drove at the Germans, taking on three players on his way into the attacking third before slipping in the unmarked Cuadrado, who was as cool as ice as he found the back of the net.
Could have - and really should have - been game over
Unfortunately for Bayern, they started to build some momentum just as half time came. With 42 minutes on the clock, Müller found space to shoot in the area, only to be denied by Gianluigi Buffon.
Pinball then ensued in the area, with a clearance bouncing off of Lewandowski and wide as the Juve goal found itself under significant threat for the first time in the evening.
Having said that, it could have and should have been game over a minute later as the Italians broke with pace and Cuadrado smashed an effort goalwards, only for Neuer to make a world class save before Lichtsteiner hit the rebound over the bar.
It was Buffon's free-kick that caused the chaos, with a simple ball over the top dealt with poorly - to say the least - by one of Europe's most feared sides. Medhi Benatia was at fault, to name names, and he will have been counting his lucky stars as he walked in at half time seeing the score read just two, not three.
Despite that being Benatia's last involvement in the game, with Juan Bernat his half time replacement, little changed after the interval too. Juve continued to press, Bayern continued to be pinned back in their own half, and the Italians remained ahead.
Morata could have increased the lead to three before the hour, with two big chances coming his way within the space of two minutes.
Both times, the Spain international was released down the left and looked to cut inside onto his favoured right to strike. On the first occasion, he was forced to go with his weaker left and could only hit the ball right down the throat of Neuer as a result. The second time, he lifted his attempt over the bar after squirming through a crowded area.
It was just down to Juve to ensure that they would not rue these missed opportunities.
Late rescue mission takes it to extra time
The conversion of these chances would have put the game to bed, but instead it was alive by the slimmest hope - one which Bayern acted upon with 73 minutes played as Lewandowski halved the deficit.
A delicious ball was whipped in by Douglas Costa, and the Poland international was at the back post to head it past Buffon and suddenly minimise the task at hand.
Despite a dreadful showing, Bayern could take the game to extra time with another goal.
Defending by the Italians suddenly became desperate. They lost their high intensity from the opening 73 minutes and were instead scrapping the ball clear in the most frantic of manners when it entered their box.
Their park the bus tactics would not withstand the time remaining either, with a stunning cross from Coman picking out Müller at the back post to head home an equaliser on the night, and on aggregate, in the first minute of stoppage time.
Momentum carries Bayern over the line
Juve needed to rediscover their earlier energy to get through this fixture, whilst it was down to Bayern to capitalise on the momentum gained from their comeback to progress.
It was the former who came out of the blocks quickest in the first half of added time though, with Mario Mandzukic - a former Bayern man himself - teeing up Lichtsteiner, who could only find the out-stretched arms of Neuer with his bending shot.
Nonetheless, Bayern assumed the driving seat role sooner rather than later as they patiently waited for an opening to appear in the back line of their organised opponents.
They didn't create many chances in the added half an hour, but they were clinical when they did as Thiago arrowed in his side's third of the night. Having come off the bench to make a difference, he did exactly that when he side-footed home Müller's lay-off on the edge of the box with less than 15 minutes to play.
Stefano Sturaro had the chance to give Juve a late hope when the ball find kindly for him in the area, but he sliced his effort well over the bar from 10 yards, allowing Bayern to keep their two goal lead.
Leonardo Bonucci ventured forward to try his luck shortly after, but he hit the ball right at Neuer as the long ball started to be lifted in the direction of the home goal.
However, they found Bayern's back line to be nowhere near as fragile as it was in the early stages of normal time, with Pep Guardiola's men able to ride their luck before applying the quality needed to progress to the last eight.