Strikes from Thomas Muller, Mario Mandzukic, as well as an own goal from Thomas Kleine broke the hosts' resistance and provided the difference in the Bavarian derby.
The Cloverleaves looked comfortable in large parts during their modern top-flight debut but their stern resistance was eventually broken by Thomas Muller before halftime. The second period followed a similar pattern but Mario Mandzukic and an own goal from Thomas Kleine added a pair of strikes to give the scoreline a glossy finish.
In a low-key start Muller controlled a throw with his chest and hooked the ball to the back post with eight minutes on the clock. Mandzukic cushioned the cross and attempted an acrobatic effort but he failed to trouble Max Grun in the Furth goal.
From then on Bayern assumed total domination of the ball and a 13th-minute free kick from Heinrich Schmidtgal that missed the target was one of very few chances for the hosts.
Despite their retention of the ball Jupp Heynckes' men were creating very little to worry their opponents. Furth was remaining compact and well organized, squeezing the space in which Die Roten could play the ball.
Nehrig and Schmidtgal were keeping things tight in the full-back positions for Furth but it was the defending from a set piece that let them down on the stroke of halftime.
Dante's header was cleared off the line by Schmidtgal but an unfortunate ricochet off of his teammate Mergin Mavraj saw the ball drop at Muller's feet and he lashed it straight into the roof of the net.
The second half began in a similar manner, with Bayern dictating the play and Furth doing its best to contain the Bavarian giants. However, it could not last as long as it did in the first half and Bayern doubled the advantage on the hour mark.
Arjen Robben was slipped in behind the defensive line and after breaking into the box he sent a right-footed shot at Grun's goal. The keeper managed to parry the effort to his side but Mandzukic was on hand to head the rebound into an empty net.
The Dutchman's performance continued to improve and after being denied by a great save from Grun with 20 minutes remaining he assisted in adding a third.
After a swift break, Robben carried the ball into the left side of the area. He looked to send it to the far side but a deflection off of Kleine diverted the ball in at the near post to cap the victory.
Claudio Pizarro missed a golden chance to add the fourth but the result still sends Bayern top of the table on goal difference and it will take on Stuttgart in its next Bundesliga fixture.