We report the observation of the 1s and 2s ground state excitons in small-offset superlattices with alternating layers of non-magnetic (ZnSe) and diluted magnetic (ZnxMn1−xSe) semiconductors. Due to the large Zeeman splitting of the electron states in ZnMnSe, valence and conduction band offsets in these structures can be tuned by external magnetic field over the scale of tens of meV. Two series of superlattices were investigated, with Mn concentrations close to 10 and 15 atomic percent, respectively. Each series consisted of five superlattices with different layer thicknesses. The 2s exciton of the ground state is observed in all samples, and exhibits a diamagnetic shift that is consistent with earlier studies. The ability to determine the energy separation between the 1s and 2s exciton states allows us to analyze our results in the framework of thefractional dimension analysis , and to extract the values of exciton binding energies for the superlattices from the analysis.