Sane had the chance to put Schalke ahead before the break but the 19-year-old's attempt from outside the box was well saved low down by Drobny.
Hamburg started the second half well, with Hunt again testing Fahrmann before Lewis Holtby went through on goal but he could not keep his lofted effort down.
However, it was Sane who would put Schalke in front on the hour mark after being played through by Leon Goretzka before the young forward rounded Drobny for his third goal this season.
Pierre-Michel Lasogga had a chance late on but a tiny flick from Goretzka put the forward off and the corner ended up bouncing off the striker's shoulder and over the bar.
The win ends Hamburg's unbeaten start as they slip to ninth in the Bundesliga table while Schalke's undefeated run continues as they join bitter rivals Borussia Dortmund on 16 points in third.