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 xIn~35% near the dot apex. We show that the observed g factor tunability is dominated by the hole, with the electron contributing only weakly. The electric-field-induced perturbation of the hole wave function is shown to impact upon the g factor via orbital angular momentum quenching, with 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.