The visitors went into the game with just one point from six previous outings and that poor run looked set to continue when Martin Harnik gave Stuttgart an 18th-minute lead.
But Jeffrey Bruma and Robert Tesche hit back in the second period to provide a welcome boost to Hamburg as they seek a replacement for Oenning.
Kick-off was delayed by 10 minutes after heavy traffic held up the 55,700-strong crowd, and Stuttgart started the stronger when the match got under way.
They took the lead when visiting keeper Jaroslav Drobny fumbled Zdravko Kuzmanovic's shot and Harnik followed up to slot the ball home.
They could have extended their lead four minutes later after Tesche's error inside his own penalty area allowed Cacau to fire in a snap-shot, which rebounded to safety off the upright.
Hamburg struggled to create chances before half-time, with Zhi Gin Lam's shot wide from the edge of the box the nearest they came.
But they were level within six minutes of the restart when Gokhan Tore whipped in a corner from the right and defender Jeffrey Bruma headed his first goal for the club.
Tore was again the provider in the 67th minute, his flag-kick this time allowing Tesche to finish from the left of the penalty area.
Stuttgart's only chance of an equaliser came two minutes from time when Kuzmanovic let fly from just inside the penalty area, but Drobny parried the ball away to ensure his side took the spoils.