The Peruvian striker grabbed his second and third goals of the 2012-13 DFB-Pokal to send the Bavarians into the next round at the expense of the lacklustre visitors
Claudio Pizarro opened the scoring with a cool drive from the edge of the box, and Arjen Robben's first goal of the season put the hosts into a commanding advantage just after the break. The Peruvian forward made it three soon after, and his Dutch colleague added insult to injury with a late fourth.
Bayern had suffered a disappointing 2-1 loss to Leverkusen at the weekend, but looked eager to get ahead early to avoid any upset here, with the returning Robben darting in off the right and firing a shot just over the bar.
However, they looked vulnerable at the back, and on eight minutes, they almost fell behind. Kostas Fortounis motored down the right, and centred the ball to Mohammadou Idrissou, who blazed over from a matter of yards when he should have buried it.
And the miss soon proved costly. Robben received a Rafinha long ball on the right wing, tickled the ball between two defenders, and Pizarro was there to drive the ball past Tobias Sippel and put his side a goal ahead.
Falling behind failed to dampen Kaiserslautern's spirits, though. Idrissou had another chance to net soon after, but Tom Starke, making his Bayern debut, palmed his shot wide, before Steven Zellner struck the woodwork with an ambitious effort from distance.
At the other end, Xherdan Shaqiri drove wide from 20 yards, before David Alaba forced a great save from Sippel with a fantastic dipping free-kick which was heading for the corner of the net.
Eventually, the match settled into more of a rhythm, with Bayern controlling possession in the centre of the park, but the hosts had a brilliant opportunity to extend their lead when Diego Contento's cross cannoned off Robben to Alaba, whose first-time volley was superbly parried by the Lautern keeper.
Bayern started a seige of the visitors' goal shortly after the break, and Sippel was forced into another save within seconds when Shaqiri tried an acrobatic volley from the edge of the box.
Seconds later, however, the hosts finally doubled their advantage. An awful Leon Jessen header fell straight to Pizarro, who nodded it onto Robben, and the Dutchman smashed it home from close range.
It was not long before the third arrived. Shaqiri swung in a free-kick from the left wing, and Pizarro, completely unmarked, headed home from the edge of the six-yard box.
Alaba then fired an ambitious effort high and wide from distance, before Pizarro missed an excellent opportunity to complete his hat-trick, taking a heavy first touch in the area and allowing Sippel to claim the ball.
It looked as though three would be enough for the visitors, but seconds before full-time, Robben made it four. Pizarro was sent clean through, only for Marc Torrejon to tackle the former Werder Bremen forward and gift the ball to the Dutchman to round off an easy night for the Bavarians.