Thomas Muller put the Bavarians in front just before half-time, with Mario Mandzukic and Arjen Robben - via a deflection - on target in the second half for Jupp Heynckes' men.
Furth rarely threatened Manuel Neuer's goal as they were given the toughest of introductions to life among Germany's elite.
Bayern came up against a predictable mass of green and white shirts as the Bundesliga new boys camped in their own half, waiting for openings to present themselves.
Mandzukic went close to opening the scoring for the visitors with an overhead kick onto the roof of the net in the ninth minute.
He had Bayern's second effort of the game, too - a weak header in the 40th minute.
Three minutes later, Bayern took the lead when Xherdan Shaqiri's corner was met by the head of defensive giant Dante. His effort was blocked on the line by Heinrich Schmidtgal, but a ricochet off team-mate Mergim Mavraj fell kindly for Muller to rifle the ball in from a yard out.
Furth had to abandon their defensive tactics and, after making a brighter start to the second half, they conceded the second goal as Bayern broke quickly.
Robben took the ball into the penalty area and his shot was pushed away by Max Grun but only to Mandzukic, who nodded in his first Bundesliga goal for his new club in the 69th minute.
Robben himself added the third, although luck was on his side as an intended cross deflected off Bernd Nehrig and looped into the goal with Grun powerless to keep it out.
Claudio Pizarro and Bastian Schweinsteiger both missed chances to make it a greater margin of victory for Bayern, who nevertheless took all three points to follow the example of defending champions Borussia Dortmund, who started the season with a 1-0 home win over Werder Bremen last night.