Both sides were missing key players for the fixture, with Dortmund particularly hindered with their regular back four all unavailable.
In a first half bereft of chances, Robert Lewandowski, who was also linked with Bayern in the transfer window, wasted the best opportunity when he fired over from close range.
Gotze's neat finish after 66 minutes finally brought the game to life before Robben and Muller also netted late on to extend Bayern's unbeaten Bundesliga record to 38 matches and open up a seven-point gap over Dortmund, last season's runners-up.
Jurgen Klopp handed a debut to centre-back Manuel Friedrich with Dortmund's usual back four missing from the starting XI. Franck Ribery was absent with a rib injury for Bayern, while Gotze was only named as a substitute against his former club.
Klopp's charges started brightly and Lewandowski should have hit the target when he fired his shot on the turn over the crossbar when unmarked in the area.
The hosts continued to look lively and Marco Reus broke clear of the defence shortly before the half-hour mark and stung the palms of Manuel Neuer with a left-foot strike.
Bayern had shown little going forward until good work on the left from Robben led to Mario Mandzukic's acrobatic effort being beaten away from Roman Weidenfeller.
The same combination then nearly handed the away side the lead against the run of play on 38 minutes, but Mandzukic failed to make clean contact with Robben's teasing cross into the six-yard area.
The first half came to an end in tempestuous circumstances as Mandzukic reacted angrily to a late challenge from Kevin Grosskreutz, with the latter lucky to avoid a red card when he appeared to aim a headbutt at the Croatian.
Neither side was able to put their stamp on the second half, until Pep Guardiola introduced Gotze to a chorus of boos from the home fans.
And the substitution paid dividends on 66 minutes when the Germany international controlled Thomas Muller's pass from the right and finished to the far corner with the outside of his boot to silence the jeers.
Dortmund pressed for an equaliser with Henrikh Mkhitaryan firing wide from close range and Reus' low drive forcing a superb reflex save from Neuer.
However, Bayern ended the game as a contest in the 85th minute when Robben broke clear of the Dortmund defence and applied a sublime chip over Weidenfeller.
And there was further cause for Bayern celebration just two minutes later. Robben sprayed the ball to Philipp Lahm and his centre was turned home by Muller as Bayern secured their first triumph in Dortmund since 2009.