Goals from Nils Petersen, Nicolas Hofler, and Vincenzo Grifo fired Freiburg into a commanding half-time lead as the visitors produced a scintillating counter-attacking display.
The Bundesliga's third-best defensive outfit then held on comfortably after the break, despite Robert Glatzel providing a late consolation for the hosts, and Freiburg will meet either RB Leipzig or Union Berlin in next month's final.
11 years after first joining the visitors, head coach Christian Streich continues to make history, with Freiburg's dreams of cup glory and a top-four Bundesliga finish remaining in their sights.
Hamburg fell behind after just 11 minutes when Petersen headed home after a goalmouth scramble, with a quick VAR check ascertaining Roland Sallai was not interfering with play.
That goal gave Freiburg a huge boost, and they doubled their lead within five minutes when Hofler's long-range strike cannoned off Sebastien Schonlau and into the roof of the net.
Hamburg almost responded when Anssi Suhonen forced a save from Mark Flekken, but were out of the contest after 35 minutes when the VAR awarded Freiburg a bizarre penalty, with Vincenzo Grifo smashing home from the spot after Moritz Heyer clumsily kicked Nico Schlotterbeck in the neck.
Suhonen had a cool finish ruled out in a marginal offside call as Hamburg's nightmare half drew to a close, before Sallai rippled the side netting with a fierce strike shortly after the break.
The 2.Bundesliga team took the game to Freiburg in the second half, with Sonny Kittel's deflected volley drawing a decent save from Flekken after 67 minutes.
Hamburg was finally rewarded for their improvement when Glatzel nodded a consolation goal after 87 minutes, and Freiburg's Ermedin Demirovic struck a post from range as the visitors claimed a historic win.