The dimensionality of the crystal structure plays a vital role in artificial heterostructures composed of different transition metal oxides. Nonlinear layer-thickness dependence of the exchange bias effect was observed in high-quality SrCuO₂/La_0.7Sr_0.3MnO (LSMO) superlattices induced in the present work by dimensional evolution. In the SCO(n)/LSMO(8) superlattices with thickness below the critical value (5 u.c.), the exchange bias effect decreased and the saturated magnetization increased with increase in SCO thickness. By contrast, the exchange bias effect increased and the saturated magnetization decreased in S(n)L(8) superlattices with thickness above the critical value. This is because the lattice SCO material underwent a breathing-like structural transformation from the planar to a chain-like structure. The results indicate the interfacial superexchange coupling mainly present in the chain-like S(n)L(8) superlattices through X-ray absorption spectroscopy and first principles calculations. This superexchange coupling generated a weak localized magnetic moment to pin the adjacent ferromagnetic layer. However, in the thicker S(n)L(8) superlattices, evolution of magnetic properties was induced by the long-range antiferromagnetic order in the planar SCO layer. Our findings demonstrate that the dimensionality driven structural variation is an effective method to manipulate the electronic reconstruction and the associated physical properties, paving a pathway for the advancement of strongly correlated materials.