Strong electrically tunable exciton g-factors are observed in individual (Ga)InAs self-assembled quantum dots and the microscopic origin of the effect is explained. Realistic eight band k.p simulations quantitatively account for our observations, simultaneously reproducing the exciton transition energy, DC Stark shift, diamagnetic shift and g-factor tunability for model dots with the measured size and a comparatively low In-composition of x(In)~35% near the dot apex. We show that the observed g-factor tunability is dominated by the hole, the electron contributing only weakly. The electric field induced perturbation of the hole wavefunction is shown to impact upon the g-factor via orbital angular momentum quenching, the change of the In:Ga composition inside the envelope function playing only a minor role. Our results provide design rules for growing self-assembled quantum dots for electrical spin manipulation via electrical g-factor modulation.