Juve were sorely exposed by the newly-crowned German champions in a 2-0 defeat in last week's first leg of their quarter-final clash.
However, Antonio Conte's side put up a better performance on Wednesday before hope faded when Mandzukic headed past Gianluigi Buffon in the 64th minute after the 'keeper had saved from Javier Martinez.
Pizarro then capitalised on some sloppy defending to beat Buffon two minutes into injury-time to leave the Italian giants with a 4-0 aggregate defeat.
Juve, unbeaten at home in the competition since a 4-1 rout by Bayern in 2009, pressed hard for an early goal but the Serie A champions ultimately came up short where it mattered.
Bayern manager Jupp Heynckes paid tribute to his side's poise on their way to an improved second half display.
"We've made the semi-finals but we were up against a really strong Juve side and the conditions for a great second leg were all there," said Heynckes.
"Juve put us under a lot of pressure in the opening half but in the second we came back into it and steadily imposed our own game."
Conte had spoken of the "gap" between this Juventus side and Bayern, who have now qualified for their third semi-final in four years having lost finals in 2010 and 2012.
The 43-year-old said: "The best side went through. They showed that last week and here tonight.
"When you're playing against one of the strongest teams in Europe it becomes difficult for everyone.
"They were simply stronger than us and we can only congratulate them."
Striker Mirko Vucinic, absent from the first leg, started alongside Fabio Quagliarella due to Sebastian Giovinco's recent knee injury and spurned a chance after just two minutes when his shot from the edge of the area lacked power.
Claudio Marchisio then shot over from Andrea Pirlo's delicate chip, while at the other end Mandzukic caused a scare when he connected with Franck Ribery's low cross as centre-back Leonardo Bonucci rallied to clear the danger.
Juve's best chance of the half came inside 20 minutes when Marchisio was fouled on the edge of the area. Pirlo's free kick was on target but Manuel Neuer pulled off a great two-handed save.
Juve were enjoying possession and momentum but just before the half hour there were no takers as Paul Pogba sent a tempting offering across the Bayern goalmouth.
Bayern were finding few opportunities, but looking dangerous when they did.
Minutes later a cross found Mandzukic unmarked and his header to Thomas Mueller left the German sending a first-time effort past Buffon's post.
Arjen Robben was then free to run into the area and found Ribery with ease, but the Frenchman's shot was blocked.
David Alaba, who stunned Buffon after just 25 seconds for Bayern's first goal last week, forced a two-handed parry from the Juve skipper on 37 minutes as the visitors finished the half with a flurry.
Juve resumed in positive fashion but when Vucinic won a free kick just outside the area Neuer easily scooped up Pirlo's low effort after it had wriggled through the defensive wall.
But barely a minute later Neuer was forced to dive smartly to his right to cover as Fabio Quagliarella's shot whizzed past the upright.
Bayern were beginning to build momentum and Robben threatened with a curling shot from 20 metres which smacked off Buffon's right-hand post.
Moments later, the Bavarians virtually sealed their last four place after Giorgio Chiellini gave away a free kick following a clash of heads with Mandzukic which left the striker on the ground.
Midfielder Bastian Schweinsteiger hit the free kick and when Buffon parried Martinez's first-timer Mandzukic was on hand to head the ball home.
The Germans should have doubled their lead shortly after when Mueller strolled in from the left only for his shot to go just wide.
Conte made three substitutions notably with Alessandro Matri replacing Quagliarella, but by then Juve's resolve had crumbled.
Robben forced a diving save out of Buffon in the closing stages, but the 'keeper was left virtually stranded in injury time as Pizarro collected Schweinsteiger's pass to hit Bayern's second.