A common origin for the Moon and Earth is required by their identical isotopic composition. However, simulations of the current giant impact hypothesis for Moon formation find that most lunar material originated from the impactor, which should have had a different isotopic signature. Previous Moon-formation studies assumed that the angular momentum after the impact was similar to that of the present day; however, Earth-mass planets are expected to have higher spin rates at the end of accretion. Here, we show that typical last giant impacts onto a fast-spinning proto-Earth can produce a Moon-forming disk derived primarily from Earth's mantle. Furthermore, we find that a faster-spinning early Earth-Moon system can lose angular momentum and reach the present state through an orbital resonance between the Sun and Moon.