The defeat left Hamburg 16th in the standings, the relegation playoff spot, and five points adrift of VfB Stuttgart above them with two games to play.
They are only one point ahead of Nuremberg, in the direct relegation places, with bottom club Eintracht Braunschweig a further point behind.
Halil Altintop swept Augsburg ahead in the seventh minute, scoring from just inside the area with a first-time effort from Paul Verhaegh's square pass, and the Turkish midfielder volleyed a second just after the half hour.
Hamburg goalkeeper Rene Adler completed a terrible first-half for the so-called Dinosaurs, the only ever-present side in the Bundesliga's 51-year history, when he allowed Andre Hahn's shot to squirm past him for the third.
Heiko Westermann headed a goal back one minute later but Hamburg, who have the Bundesliga's worst defensive record with 68 goals conceded, had left themselves with far too much to do in the second half.
"Our best chances came from set-pieces, but from open play we didn't do enough," Hamburg coach Mirko Slomka told Sky.
"We had to respond to falling behind, but instead we kept giving the ball away sloppily."
In Sunday's other match, Schalke lost 1-0 at home to Borussia Monchengladbach.
- Sports & Recreation
- Eintracht Braunschweig
- VfB Stuttgart
- Paul Verhaegh