The defeat dented Dortmund's chances of returning to the summit of the German top-flight standings, while Huub Stevens confirmed their safety from relegation with 30 points from 31 matches.
Sane fired Schalke ahead in the 28th minute after Daniel Caligiuri had cancelled out Mario Gotze's opener at the Signal-Iduna-Park.
The effort was Sane's second goal in his debut campaign with the Veltins-Arena outfit.
Second half efforts from Caligiuri and Bastian Oczipka sealed maximum points for the visitors as Axel Witsel's 84th-minute goal stood as a consolation for Lucien Favre's side.
Dortmund are now a point adrift of league leaders Bayern Munich - who have a game in hand - with 69 points from 31 games.