Germany star Thomas Mueller was Barca's chief tormentor with goals in each half while striker Mario Gomez and Dutch wing Arjen Robben also netted to give Bayern a huge advantage for the return leg at the Camp Nou on May 1.
The stunning performance came at the end of a remarkable day even by Bayern's standards. First they announced they had signed young German star Mario Goetze from arch-rivals Borussia Dortmund for a reported 37 million euros.
Then it was reported that in March club president Uli Hoeness caught up in a tax evasion scandal, was arrested and released on bail after a police raid on his home.
But he was at the Allianz Arena to witness an incredible night for the hosts. Barcelona's Argentinian superstar Lionel Messi started, despite struggling with a hamstring injury since the start of the month, but the world player of the year was a shadow of his usual self.
Bayern had an early penalty appeal turned down, but the hosts went ahead when Robben's lobbed cross from a corner was headed across goal by centre-back Dante for Mueller to place his header inside the far post on 24 minutes.
Dante came to his side's rescue at the other end five minutes later after Pedro Rodríguez's cross from the right was heading straight for Messi at the back post, but the Brazilian got a toe to the ball.
Bayern doubled their lead in the 49th minute as Gomez struck with his sixth goal in his last three games. Again, the Barcelona defence failed to deal with a corner as Robben's ball from the right was nodded down when Mueller rose above Daniel Alves for Gomez to volley home on 49 minutes.
Robben, who had been a constant menace to the Barca defence, grabbed the third on 73 minutes when the Dutchman skipped past Jordi Alba on the right, then slid his shot past Victor Valdes from a tight angle.
However, the Barcelona players were furious and protested to the referee as Alba had been body-checked by Mueller as he chased back after Robben inside the box.
With the Barcelona defence in a shambles, Bayern grabbed the fourth on 82 minutes when Franck Ribery and David Alaba combined on the left for the latter to whip in a low ball which Mueller slid into the net for his second.
Despite the advantage, Bayern know how tough life could be at the Camp Nou, having lost 4-0 on their last Champions League visit to Barcelona in the 2009 quarter-final, first leg which they went on to lose 5-1 on aggregate.