The Magnetoelectric (ME) effect in solids is a prominent cross correlation phenomenon, in which the electric field (E) controls the magnetization (M) and the magnetic field (H) controls the electric polarization (P). A rich variety of ME effects and their potential in practical applications have been investigated so far within the transition-metal compounds. Here, we report a possible way to realize the ME effect in organic molecular solids, in which two molecules build a dimer unit aligned on a lattice site. The linear ME effect is predicted in a long-range ordered state of spins and electric dipoles, as well as in a disordered state. One key of the ME effect is a hidden ferroic order of the spin-charge composite object. We provide a new guiding principle of the ME effect in materials without transition-metal elements, which may lead to flexible and lightweight multifunctional materials.