Klaas-Jan Huntelaar, who has been sidelined since August with a knee injury, marked his return to first-team action by heading the visitors into a 1-0 lead from Jefferson Farfan's cross on 34 minutes.
Farfan added a second within eight minutes of the second half, beating Hamburg goalkeeper Jaroslav Drobny to a 50-50 ball before slotting into an empty net.
The Peru international turned provider once again, producing a cut-back to the awaiting Max Meyer, who swept home Schalke's third of the evening.
The hosts should have bagged a consolation with 10 minutes remaining, as Milan Badelj saw his bid to reduce the arrears rebuffed by the post.
Bert van Marwijk's men remain in the relegation zone in Germany's top flight, while Jens Keller's keep the pressure on Borussia Monchengladbach in the race for a Champions League spot.