Goetze stabbed the ball home in the 66th minute, only 10 minutes after coming on as a substitute to a resounding chorus of jeers, as the Bavarians enjoyed their first league win over Dortmund since February 2010.
The 21-year-old, whose transfer stunned Borussia fans when it was confirmed last April for 37 million euros ($50.04 million), did not celebrate his goal.
Arjen Robben sealed the win with a typical breakaway effort with five minutes left and Thomas Mueller sidefooted a third to complete a demoralising evening for Dortmund as their makeshift defence collapsed.
Dortmund were left to rue an early miss by Robert Lewandowski, who fired wide from six metres after only two minutes, and a brilliant save by Bayern goalkeeper Manuel Neuer to deny Marco Reus with the score 1-0.
The win took leaders and champions Bayern, who have 35 points, seven clear of third-placed Dortmund.
Bayer Leverkusen are sandwiched inbetween on 31 points after winning 1-0 at Hertha Berlin with Stefan Kiessling's 13th goal of the season in all competitions.
Bayern, who had not won any of their last six meetings with Dortmund, had plenty of early possession but spent most of the team passing the ball sideways, unable to play their way out of defence.
After Lewandowski's miss, Jakub Blaszczykowski wasted a chance for Dortmund when he dallied over his shot and Reus had a powerful drive parried by Neuer as the hosts looked menacing.
Dortmund had their entire first-choice back line of Lukasz Piszczek, Mats Hummels, Neven Subotic and Marcel Schmelzer out injured but were in little danger in the first half hour.
Bayern's best first-half chance fell to Mario Mandzukic who just failed to make proper contact after Robben rolled the ball across the penalty area following a break down the left.
Dortmund were again close early in the second half when Kevin Grosskreutz chipped the ball in from the right, Lewandowski and Neuer missed it and the ball bounced agonisingly wide.
The game swung Bayern's way when Mueller found Goetze and he controlled the ball with his left foot and immediately stabbed it home with his right.
Reus thought he had levelled with a shot which went through Dante's legs only for Neuer to make a superb one-handed save at the foot of the post.
Bayern finished off the game when Dante's long, crossfield pass found Robben and he burst down the left and lofted the ball over keeper Roman Weidenfeller.
Mueller gave the scoreline a flattering look when he turned in Philipp Lahm's pass from inside the area.
"The size of the score is irrelevent, we played well until the last eight minutes and didn't get a reward," Weidenfeller told Sky Sports Deutschland.
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