A series of new magnetically-active coordination compounds comprising binuclear mixed-ligand complexes of cobalt bis-diketonates with acene linkers functionalized by two redox-active o-quinone moieties has been designed by means of density functional theory (DFT UB3LYP*/6-311++G(d,p)) calculations of their electronic structure, energy characteristics and magnetic properties. Two types of redox-active ligands include those with an acene linker bridging two o-benzoquinone fragments and the ligands containing an integrated π-conjugated system formed by annulation of o-quinone rings to the polycyclic core. The calculations reveal the dependence of spin density distribution in the compounds under study on the type of ligand. The considered binuclear Co^II diketonate adducts manifest the capability of undergoing one- and two-step spin transitions induced by intramolecular electron transfers between the metal ions and the ligand system. An increase in the number of condensed rings of the acene linker promotes the stabilization of the biradicaloid state of the linker and enhances exchange interactions between all paramagnetic centers of the complexes giving rise to the formation of a flexible spin coupling network. The predicted unusual magnetic properties make the binuclear cobalt complexes with di-o-quinone ligand containing acene linker groups the promising building blocks for molecular electronic and spintronic devices.