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A conceptual framework for the origin of the Moon must explain both the chemical and the mechanical characteristics of the Earth-Moon system to be viable. The classic concept of an oblique giant impact explains the large angular momentum and the lack of a large iron-rich core to the Moon, but in this scenario it is difficult to explain the similarity in the isotopic compositions of the Earth and Moon without violating the angular momentum constraint. Here we propose that a giant, solid impactor hit the proto-Earth while it was covered with a magma ocean, under the conventional collision conditions. We perform density-independent smoothed particle hydrodynamic collision simulations with an equation of state appropriate for molten silicates. These calculations demonstrate that, because of the large difference in shock heating between silicate melts and solids (rocks), a substantial fraction of the ejected, Moon-forming material is derived from the magma ocean, even in a highly oblique collision. We show that this model reconciles the compositional similarities and differences between the Moon and Earth while satisfying the angular momentum constraint.