A Lionel Messi double and Neymar's strike at Camp Nou last Wednesday had given the Liga leaders a 3-0 lead after the first encounter and Bayern were unable to complete a remarkable fightback six days later despite a spirited display.
Mehdi Benatia's early header gave Bayern hope, but Pep Guardiola described his former club Barca as the best counter-attacking team in the world ahead of the game and they lived up to that billing at the Allianz Arena.
Luis Suarez laid on two goals for Neymar - who now has 37 for the season and had scored in his last seven matches - in the first half and, although Robert Lewandowski and Thomas Muller gave the Bundesliga champions victory on the night, the job had been done by Barca in the first leg.
It was Barca's first defeat since February and they had not conceded for over 10 hours prior to Tuesday's game, but they remain on course for a famous treble in Luis Enrique's first season in charge, while Bayern have crashed out in the semi-finals two years running under Guardiola.
Luis Enrique had stated he did not want a "crazy" clash in Bavaria, but the Barca coach must have been fearing the worst when Benatia gave Guardiola's side the lead on the night after only seven minutes.
The defender punished some slack marking by meeting Xabi Alonso's corner with a header that Marc-Andre ter Stegen got a hand to, but failed to keep out.
Bayern never looked capable of keeping the Liga leaders' potent attacking trio quiet, though, and they combined with great effect for a deserved equaliser eight minutes later.
Messi found Suarez with an incisive pass and the Uruguay striker showed great awareness to square the ball for Neymar to tap into the empty net after Manuel Neuer had come out to narrow the angle.
Ter Stegen then showed outstanding agility to keep out Muller's header, but Bayern were made to pay for more poor defending when Neymar struck again after 29 minutes.
The lively Suarez raced away all too easily and was once again allowed the time to unselfishly lay the ball across goal for Neymar, who controlled on his chest and shot inside the near post with his right foot.
Lewandowski really ought to have equalised when he found space in the penalty area, but his initial shot was brilliantly saved by Ter Stegen and the German goalkeeper rose to his feet sharply to scramble back in order to just about prevent the whole ball from crossing the line.
Pedro replaced the impressive Suarez at half-time and Lewandowski made no mistake with a fine finish to make it 2-2 on the night just before the hour-mark.
The Poland striker showed lovely footwork to beat Javier Mascherano just inside the area before showing great composure into the bottom-right corner of the net.
Muller grabbed a winner 16 minutes from time with a lovely first-time finish, but the damage had been done in Catalonia.
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