Bundesliga Review: Bayern march on as Dortmund struggle
It proved to be a torrid afternoon for Cologne midfielder Daniel Halfar, whose mistake let in Thomas Muller to tee up Mario Gotze for the opener after 19 minutes.
Things went from bad to worse for Halfar after half-time as he scored an own goal in the 66th minute under pressure from David Alaba.
In contrast to Bayern, Borussia Dortmund's indifferent start to the campaign continued with a 2-1 loss at Schalke that leaves them 10th in the standings.
Joel Matip powered home from a 10th-minute corner before Eric Maxim Choupo-Moting capitalised on some sloppy 23rd-minute defending to send Schalke well on their way to Revierderby bragging rights.
In-form forward Pierre-Emerick Aubameyang reduced the deficit and Schalke goalkeeper Ralf Fahrmann then displayed inspired form to condemn Dortmund to their third defeat in six league games.
Borussia Monchengladbach moved into second place, two points behind Bayern, as they continued their own unbeaten start to the campaign with a 2-1 win at SC Paderborn.
Patrick Herrmann capped a lightning counter-attack for Monchengladbach to open the scoring and Raffael had them two goals to the good after just 14 minutes.
To their credit, Paderborn - who were beaten 4-0 by Bayern in midweek - refused to fold and Jens Wemmer's 70th-minute strike set up a grandstand finish, but the visitors held on.
Bayer Leverkusen drop down to third after they failed to score for the first time this season in a goalless draw at Freiburg.
Leverkusen defender Emir Spahic was sent off for a second bookable offence shortly before the half-hour, while the hosts had Pavel Krmas red-carded in the 78th minute.
Freiburg slip into the bottom three after Stuttgart followed a spirited midweek draw against Dortmund with a first win of the league season, a 1-0 victory at home to Hannover.
Daniel Schwaab netted his first goal in 134 Bundesliga outings in the 69th minute to send the Mercedes-Benz Arena into raptures.
Elsewhere, Wolfsburg got back to winning ways with a 2-1 victory at home to second bottom Werder Bremen.