Right, drag scales with the square, but you're also going into that drag at a higher velocity, so power scales with the cube.
Or to break it out at more length, work is force times distance, right, and power is work over time. If it takes X amount of work to cover a distance at a given speed, it takes 4X to do it in half the time, since, as you point out, the drag (force) scales with the square of velocity. But you also accomplished that work in half the time, so the power required is 8 times the original. Make sense?
1: http://en.m.wikipedia.org/wiki/Drag_equation