Pep Guardiola's side may have been unspectacular in the opening four games of their title defence, yet Bayern never looked like losing at the Allianz Arena against Paderborn, as Gotze's double, Robert Lewandowski's half-volley and Thomas Muller's close-range finish late on secured victory and top spot in the table.
Paderborn, competing in the Bundesliga for the first time, sat top on goal difference ahead of Bayern prior to Tuesday's game after an encouraging start to the campaign, but they were well beaten by the champions.
Gotze gave Bayern the lead after just eight minutes, steering home following a fine flick from Muller.
Lewandowski hit back at his critics with a 25-yard effort in the 14th minute and Gotze netted his second 12 minutes from time before Muller applied a neat finish to Arjen Robben's 85th-minute cross to end Paderborn's unbeaten start to Bundesliga life.
Guardiola threatened to ban his players from celebrating Oktoberfest if they failed to beat Paderborn, but Bayern never looked like slipping up as Gotze broke the deadlock early on.
Robben, back after missing Saturday's goalless draw with Hamburg, played Muller into the area and the German forward produced a delightful backheel for Gotze, who stroked a first-time effort into the net.
Gotze fired a tame effort at Lukas Kruse as he went in search of a second, but Bayern did make it 2-0 after 14 minutes.
Lewandowski produced a fine first touch 25 yards out, teeing the ball up perfectly for a half-volley that Kruse failed to keep it out.
The hosts continued to attack, as Lewandowski, Gotze and David Alaba all saw chances go begging, while Idir Ouali prodded straight at Manuel Neuer on the counter-attack for Paderborn's best chance 24 minutes in.
Gotze nearly added a third just before the break, as his low drive from the edge of the area deflected agonisingly wide.
Bayern started the second half in a far more relaxed fashion and that seemingly presented Paderborn with the opportunity to fight their way back into the contest.
The visitors crafted a good opening in the 58th minute as Rafinha cheaply surrendered possession and allowed Ouali to deliver a teasing cross from the left, but Xabi Alonso crucially poked it behind for a corner.
Bayern needed Neuer at his best in the 70th minute as Suleyman Koc ran at Alaba and sent a stinging effort towards the bottom-right corner.
Any hope of a late comeback was ended 11 minutes from time, though, as Gotze smashed in from close range following a clever Robben run down the right.
And the Dutch winger then played a decisive role again in Bayern's fourth.
The former Chelsea man darted past his man and fired a low ball in towards the front post to allow Muller to nip in to divert past the helpless Kruse.
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