Three new hetero-metallic CuII-LnIII complexes [(CuL)Gd(NO3)3(CH3OH)]n (1), [(CuL)Tb(NO3)3(H2O)]·[CuL] (2) and [(CuL)Dy(NO3)3(H2O)]·[CuL] (3) have been synthesized using a mono-nuclear Cu(II) complex, [CuL], of an unsymmetrically di-condensed N2O3 donor Schiff base ligand, N-(3-methoxysalicylidene)-N-(salicylidene)-1,2-ethylenediamine (H2L). Single crystal X-ray crystallography revealed that complex 1 is a nitrate bridged 1D chain of dinuclear Cu(II)-Gd(III) units whereas in 2 and 3, the dinuclear Cu(II)-Ln(III) units are co-crystallized with a [CuL] unit. The Ln(III) centers are nine coordinated with the geometry of a spherical capped square antiprism for Gd and spherical tricapped trigonal prism for Tb and Dy. The geometry of the Cu(II) center is distorted octahedral for complex 1 and distorted square planar for complexes 2 and 3. Temperature-dependent molar magnetic susceptibility measurements in 1-3 revealed the presence of overall ferromagnetic coupling between the Cu(II) and Ln(III) centers. Notably, field induced single-molecule magnet behavior was witnessed in the Tb(III) derivative (2). The ab initio calculations indicated that upon application of an external magnetic field, the tunneling in the ground state of complex 2 gets reduced and thereby field-induced SMM behaviour is observed. Besides, in the case of complex 1, BS-DFT calculations were carried out to gain further insights into the magnetic exchange coupling interactions between the Cu(II) and Gd(III) centers.