Controlling the growth of complex relaxor ferroelectric thin films and understanding the relationship between biaxial strain-structural domain characteristics are desirable for designing materials with a high electromechanical response. For this purpose, epitaxial thin films free of extended defects and secondary phases are urgently needed. Here, we used optimized growth parameters and target compositions to obtain epitaxial (40-45 nm) 0.67Pb(Mg_1/3Nb_2/3)O₃-0.33PbTiO₃/(20 nm) SrRuO₃ (PMN-33PT/SRO) heterostructures using pulsed-laser deposition (PLD) on singly terminated SrTiO₃ (STO) and ReScO₃ (RSO) substrates with Re = Dy, Tb, Gd, Sm, and Nd. In situ reflection high-energy electron diffraction (RHEED) and high-resolution X-ray diffraction (HR-XRD) analysis confirmed high-quality and single-phase thin films with smooth 2D surfaces. High-resolution scanning transmission electron microscopy (HR-STEM) revealed sharp interfaces and homogeneous strain further confirming the epitaxial cube-on-cube growth mode of the PMN-33PT/SRO heterostructures. The combined XRD reciprocal space maps (RSMs) and piezoresponse force microscopy (PFM) analysis revealed that the domain structure of the PMN-33PT heterostructures is sensitive to the applied compressive strain. From the RSM patterns, an evolution from a butterfly-shaped diffraction pattern for mildly strained PMN-33PT layers, which is evidence of stabilization of relaxor domains, to disc-shaped diffraction patterns for high compressive strains with a highly distorted tetragonal structure, is observed. The PFM amplitude and phase of the PMN-33PT thin films confirmed the relaxor-like for a strain state below ∼1.13%, while for higher compressive strain (∼1.9%) the irregularly shaped and poled ferroelectric domains were observed. Interestingly, the PFM phase hysteresis loops of the PMN-33PT heterostructures grown on the SSO substrates (strain state of ∼0.8%) exhibited an enhanced coercive field which is about two times larger than that of the thin films grown on GSO and NSO substrates. The obtained results show that epitaxial strain engineering could serve as an effective approach for tailoring and enhancing the functional properties in relaxor ferroelectrics.