Rafael van der Vaart put Hamburg on the way to victory before Lasogga struck three times in seven minutes to put the result beyond any doubt.
With the home fans already leaving the Frankenstadion en masse, Tolgay Arslan added a fifth to complete Hamburg's first away win of the season and only their second overall, while heaping the pressure on home coach Michael Wiesinger.
Freiburg are still looking for their first league win of the season after a late goal from substitute Nicolas Hofler earned them a 1-1 draw at home to Eintracht Frankfurt.
An own goal from Christian Gunter after 64 minutes had given the visitors the lead before Hofler equalised with five minutes to go.
The visitors had substitute Johannes Flum sent off in stoppage time after he was shown a second yellow card but held out to claim a point.