Bayern got off to a bad start at second-placed Leipzig when Marcel Sabitzer opened the scoring in the second minute.
The 28-year-old's first strike was his 150th goal in the Bundesliga - 76 of them for Bayern - and for good measure he made it 151 in German league football with his second of the day.
Carlo Ancelotti's star-studded Bayern, who were already confirmed as Bundesliga champions for the fifth straight year, were 4-3 down on 90 minutes until David Alaba drilled home a free-kick a minute later.
Injury-prone Germany defender Holger Badstuber is looking for a new club after employers Bayern Munich said on Friday they will not extend his contract following his loan to Schalke.
Bayern Munich winger Douglas Costa is reportedly a target for Juventus this summer, with the Serie A side said to be keen to add further depth to their squad.
The 33-year-old Robben proved he had the legs to sprint clear of the tiring home defence as he capped a superb run with the winning goal in the top-of-the-table clash. Naturally we expect more of him with his qualities, but it is not easy for a young player to adjust to a new country, a new style of play. Leipzig were very strong at the beginning, we were better towards the end.
Liverpool's defensive woes have been much publicised this season, and although their first choice pairing of Dejan Lovren and Joel Matip are playing at the moment, they are still conceding sloppy goals.
Guido Burgstaller's header midway through the first half left Hamburg desperately searching for an equalizer, but it didn't come until Lasogga scored in injury time.
Leipzig will be dissapointed but won't be too annoyed as they should still compete in next season's Champions League (once ownership issues are clarified).
"Congratulations Bayern for this mad victory and congrats to those who were in the stadium to witness it", said Leipzig coach Ralph Hasenhuettl. "For such young players it is often hard to get used to", he added.
Alfred Finnbogason put the home side in front minutes before Pierre-Emerick Aubameyang equalized for Dortmund, which stayed third on goal difference from Hoffenheim.
Three goals from the 84th minute restored Bayern's pride.
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