Artjoms Rudnevs scored both goals while the crowd joined the so-called '12:12 protests' against new security measures proposed by the German league.
Fans stayed silent for the first 12 minutes 12 seconds before bursting into life.
They counted down the final few seconds, then threw ticker tape and began singing and waving flags. Until then the game had been eerily quiet with shouting by the players clearly audible.
The protests began last weekend at various Bundesliga matches and continue until the clubs meet on Wednesday to discuss the proposals.
Hoffenheim, who sacked Babbel on Monday after he had been in charge for nine months, showed why they have the worst defensive record in the league as they fell behind in the 27th minute.
Latvia forward Rudnevs was left unmarked just inside the area as he met Dennis Diekmeier's cross with a header which caught goalkeeper Koen Casteels badly positioned and looped into the goal.
Tolgay Arslan hit the crossbar before halftime for Hamburg who then continued to dominate after the break and extended their lead when Rudnevs scored following a free kick, despite failing to connect properly with the ball.
It was his sixth league goal of the season and the 38th conceded by Hoffenheim who are third from bottom with 12 points from 16 games.
Hamburg, missing Dutchman Rafael van der Vaart through injury, climbed to sixth on 24.