Shinji Kagawa and Mats Hummels were also on the scoresheet as the Poland international's red-hot form propelled BVB to their first DFB-Pokal title in 23 years
Shinji Kagawa put BVB ahead early on but Arjen Robben levelled the game with a penalty midway through the first half. Mats Hummels restored Dortmund's lead before Robert Lewandowski increased it in stoppage time.
The Pole then added his second with a thunderous finish just before the hour mark that seemed to put the trophy well beyond Bayern's reach. Franck Ribery ignited a late surge for Die Roten with an excellent solo effort but Lewandowski completed his hat-trick in the dying moments to put the game beyond Jupp Heynckes' men.
The Bavarians started the match promisingly but fell behind to Dortmund's first attack with just three minutes played.
After giving away possession in his own half Luiz Gustavo then sold Manuel Neuer short with a weak back pass. Jakub 'Kuba' Blaszczykowski managed to steal in and square the ball for Kagawa to slot the opening goal into an empty net, no doubt pleasing Sir Alex Ferguson as the Manchester United manager watched from the stands.
The Bavarians went close through Mario Gomez soon after and their perseverance eventually paid off when the striker poked the ball away from Weidenfeller and was brought down for a penalty. The keeper was only booked, but Robben stepped up and converted the spot-kick with 25 minutes on the clock.
The game was being evenly contested and after Hummels made a goal-line block from Philipp Lahm's shot, he made himself a hero at the other end moments before the break. Jerome Boateng fouled Kuba in the area and the centre-back stepped up to face Neuer from the spot. The Bayern keeper got a hand to the ball but could not stop it from finding the net.
Dortmund then managed to add another goal in first-half stoppage time when Kagawa played a fantastic reverse pass into Lewandowski. The Pole fired a low shot that found a gap between the keeper's legs to double BVB's lead.
The second half continued to go from end-to-end with Dortmund proving very dangerous on the break and after a number of let offs, Jurgen Klopp's side found their fourth goal.
Kagawa fed Kevin Grosskreutz, who played a perfect pass to an unmarked Lewandowski at the back post and the Pole slammed an emphatic finish past Neuer.
The nature of the game was encapsulated just before the 70-minute mark. Some intricate play in the Bayern box allowed Lewandowski to get a shot away but Neuer held the advantage as he saved from an acute angle. The Bavarians then took the ball up the other end and almost pulled one back but Gomez's header rattled the post.
Bayern did manage to reduce the deficit with a superb individual effort from Ribery. He showed great strength and control as he carried the ball across the box from the right before rifling an effort into the bottom corner.
Lewandowski completed his hat-trick when he headed home from Lukas Piszczek's cross after the right-back capitalised on a spilled ball by Neuer.
It was a fantastic win for Dortmund as Klopp secured the club's first DFB-Pokal title in 23 years to complete their first-ever league and cup double. Bayern meanwhile must now focus on the Champions League final against Chelsea next weekend which offers their only remaining chance of silverware this season.