Late first-half goals from Harry Kane and Thomas Muller turned the tie on its head, just moments after the first-leg hero for Lazio Ciro Immobile had missed a golden chance at the other end for the Biancocelesti.
Kane added his second after the interval as Bayern cruised to a relatively comfortable victory in what was a high-pressure game for everybody involved at the club.
This is the first time in eight attempts that Bayern have progressed after losing the first leg of a Champions League knockout tie, but it is the third successive occasion in which they won a last 16 encounter despite trailing after the first leg.
It extends Bayern's fine run of making it past the last 16 for the 12th time in 13 seasons, and in doing so, they have made it seven wins in a row against Italian clubs at home.
Pressure will settle slightly on manager Thomas Tuchel for now, after coming into the game with just one win from their last five, but success in this competition is the only likely way this season can be salvaged for Bayern.
For Lazio, it extends their miserable run away in Germany in this competition, as it is now five trips without a win.
The defeat also means for just the second time in 14 previous European ties, Lazio have been eliminated despite winning the first leg.
Many expected Maurizio Sarri to shut up shop and park the bus in Bavaria this evening, but Lazio were expansive and adventurous in possession early on, committing many players forward whenever possible, and it was Matteo Guendouzi who had the first opening inside five minutes, but fired wide.
Bayern, as expected, were the dominant force in the opening stages, and Leroy Sane, Jamal Musiala and Kane all had opportunities in the opening 15 minutes that went begging.
Lazio had to be brave in defence, and a double block from Mario Gila and Adam Marusic denied Musiala and Raphael Guerreiro in quick succession as tension grew around the Allianz.
Bayern coach Tuchel was visibly frustrated when referee Slavko Vincic slipped and inadvertently intercepted a Muller pass in the final third, with nothing as of yet going right for the hosts.
There was a huge let off for Bayern just moments later though, as Mattia Zaccagni's cross, which nicked off Matthijs de Ligt, found Immobile at the back post, but the 34-year-old was unable to steer his header on target despite being completely unmarked in the penalty area.
Tuchel's men took advantage almost instantly, as a fine move saw Muller knock a header down for Guerreiro, and even thought the left-back scuffed his effort, Kane was on hand to divert it goalwards, as the ball squirmed agonisingly underneath the body of Ivan Provedel in the Lazio goal.
Bayern ramped up the pressure following the opening goal, and shortly after Musiala saw his effort roll the wrong side of the post, the host did then take the lead in the tie.
After an outswinging corner found its way to De Ligt at the back post in acres of space, his acrobatic volley was nodded in by Muller inside the six-yard box in typical Muller fashion, with replays showing he had been kept onside by Felipe Anderson.
From there on, Lazio struggled to get a foothold in the game, and despite Sarri ringing the changes early in the second half, the game soon got away from the Biancocelesti.
Approaching the midway point of the second half, Sane was allowed to step inside and get a shot away, and after Provedel parried his effort out, the man on hand in the right place at the right time was Kane to net his second of the night, and put the game beyond the visitors once and for all.
Confidence grew within the Bavarian ranks, and they showed no mercy on the faltering Lazio backline, as a counter-attack allowed Musiala to feed Muller, whose effort cannoned off the post thanks to an important hand by Provedel.
There was collective gasp around the Allianz shortly after though, when Luis Alberto received possession in the area, and his strike took an important deflection off Musiala which prevented it from troubling Manuel Neuer in the Bayern goal.
Bayern saw the game out very comfortably though as Lazio barely laid a glove on the six-time winners, who progressed 3-1 on aggregate, with Neuer not needing to make a save for the entire evening.
With the league title slipping away from Bayern, they face relegation candidates Mainz at the weekend, while Lazio must focus on an unlikely push for the top four when they host Udinese next Monday.