The Turkey international fired the hosts ahead after just six minutes, receiving a pass from his teammate Paul Verhaegh before finding the top corner from the edge of the box.
Heiko Westermann spurned a good chance to level matters, before Altintop made him pay the ultimate price for the miss by doubling Augsburg's lead shortly after the half-hour mark.
It was Hahn himself who added the third three minutes before the break, rifling his shot beyond the reach of Adler from 25 yards as Markus Weinzierl's side closed in on an emphatic half-time lead.
However, the visitors threw themselves a lifeline over staging a second-half comeback as Westermann made amends for his earlier miss by reducing the deficit on the stroke of the interval.
Nevertheless, the two-goal deficit proved insurmountable for Hamburg as the hosts closed out the encounter in comfortable fashion.