The day's events were overshadowed though by the suicide attempt from referee Babak Rafati that forced the postponement of the Cologne v Mainz match.
The 41-year-old failed to show up for the fixture and Cologne police said he was found by his assistants in his hotel room bath with his wrists slit.
"Rafati is out of danger," German FA President Theo Zwanziger told reporters. "He is in a stable condition."
Germany international Gotze scored for Dortmund midway through the second half to earn the champions their sixth win in seven league games and lift them to second place on 26 points, two behind Bayern.
"That was really hard work," Dortmund coach Jurgen Klopp told reporters. "That is how you must play if you want to get something out of such a game, 95 percent is not enough."
Borussia Monchengladbach are third, also on 26 points, after crushing Werder Bremen 5-0 with a Marco Reus hat-trick.
Even with Dutch forward Arjen Robben making his first Bayern appearance since groin surgery last month, the much-hyped game in Munich failed to live up to expectations in an error-ridden first half.
Dortmund, having worked hard to keep Bundesliga top scorer Mario Gomez at bay, came agonisingly close early in the second period when Shinji Kagawa was denied by keeper Manuel Neuer and Gotze volleyed the rebound wide.
Gotze quickly made amends, benefiting from a deflection and bad defending from Jerome Boateng to fire in from close range in the 65th minute.
Bayern lacked the commanding midfield presence of the injured Bastian Schweinsteiger and had few ideas about how to crack open Dortmund's tight defence.
Franck Ribery missed Bayern's best chance when he raced clear but keeper Roman Weidenfeller stood his ground to protect his side's lead 12 minutes from time.
"You have to build up more pressure if you play against Dortmund than we did today," said Bayern coach Jupp Heynckes. "We did not understand how to up the tempo and create chances."
SPARKLING REUS
Forward Reus inspired Gladbach to their big win over Werder as they continued a spectacular run of form after narrowly avoiding the drop last season.
The 22-year-old Germany international, who has now scored at least two goals in each of his last three league games, struck three times after Patrick Herrmann had given the hosts a 16th minute lead.
Juan Arango completed the rout in the 53rd.
Dutch striker Klaas-Jan Huntelaar netted twice and Raul and Lewis Holtby were also on target to give Schalke 04 a 4-0 win over struggling Nuremberg.
Schalke's third victory in four games put them fourth on 25 points.
VfL Wolfsburg won 4-1 against Hannover 96 to offer coach Felix Magath, who has suspended three of his players for a lack of commitment, something to smile about.
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