Kevin Grosskreutz and Mario Gotze, two of the revelations as Dortmund's previously underrated squad won last season's title, scored in the first half as the hosts started where they left off in May.
Grosskreutz, 23, added the third early in the second half as a packed crowd of nearly 80,000 were treated to some superb attacking football from the hosts.
"We played opponents today who were simply a class above us," said Hamburg captain Heiko Westermann.
"Borussia Dortmund today play in a different league, their passing of the ball was extremely good," he said.
Dortmund shrugged off the absences of Nuri Sahin, last season's playmaker who has been sold to Real Madrid, and injured Paraguay striker Lucas Barrios, their top scorer last season, as they tore the Hamburg defence apart.
Kagawa hit the post and then fired wide after Grosskreutz pulled the ball back as Jurgen Klopp's team played some delightful one-touch football capped by a brilliant second goal in the 28th minute.
Hamburg's woes deepened when Peru striker Paolo Guerrero, top scorer in last month's Copa America with five goals, limped off injured before the break.
Tesche fired in after a goalmouth scramble and Hamburg, with a late flourish, nearly scored again in the next attack when Marcel Jansen's shot was just saved by Roman Weidenfeller.
