Hamburg came into the game having won back-to-back matches for the first time in the campaign, but their problems began when Muller converted a 23rd-minute penalty.
Gotze doubled Bayern's advantage and a brilliant long-range strike from Robben made it 3-0 at half-time.
The Netherlands winger was on target again in the 47th minute and Robert Lewandowski got in on the act a minute after Muller sumptuously curled home the fifth.
Franck Ribery came off the bench to move the scoreline on to 7-0 and Hamburg's humiliation was complete when Gotze concluded the rout two minutes from time.
Pep Guardiola's team remain eight points ahead of Wolfsburg, who completed an astonishing 5-4 win at Bayer Leverkusen thanks to striker Bas Dost netting his fourth goal of the contest deep into stoppage time.
Dost, who helped himself to a double when Dieter Hecking's men stunned Bayern 4-1 last month, scored either side of Naldo's 17th-minute contribution to surge Wolfsburg into a 3-0 lead inside half an hour at the BayArena.
But by the time Dost brought up a hat-trick, Leverkusen forward Son Heung-min had two goals on the way to a treble of his own.
In the 72nd minute - four minutes after Son's third - Karim Bellarabi hauled Leverkusen level before the game took another twist.
Home centre-back Emir Spahic collected a second booking with eight minutes of normal time remaining and, with the last, remarkable act of a pulsating affair, Dost flicked home at the near post steal the points.
Borussia Monchengladbach capitalised on slip-ups by rival UEFA Champions League qualification candidates Schalke and Augsburg to move into third courtesy of some more late drama.
Cologne appeared set to leave Borussia-Park with a share of the spoils, but Granit Xhaka scored the only goal of the match in the first minute of injury time.
Schalke were on the wrong end of a 1-0 scoreline at Eintracht Frankfurt, Lucas Piazon settling matter for the hosts after 64 minutes.
Werder Bremen moved their resurgence to nine wins from 13 matches by beating Augsburg 3-2.
Viktor Skripnik's team lie behind Hoffenheim on goal difference after the seventh-placed side secured a 2-1 home win over basement boys Stuttgart.
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