Additive manufacturing (AM) enables the realization of architected lattice structures with complex geometries and tunable mechanical properties, providing new opportunities for the design of auxetic metamaterials exhibiting a negative Poisson’s ratio (NPR). In this contribution, a novel three-dimensional auxetic unit cell is proposed and systematically analyzed within a computational mechanics framework. An analytical model based on the strain energy method and Castigliano’s theorem is developed to predict the effective elastic modulus and Poisson’s ratio in all three principal directions. The formulation explicitly accounts for bending, shear, and axial deformation modes, as well as strut overlap effects at the joints, enabling accurate, geometry-dependent property predictions. The analytical expressions are validated against finite element simulations conducted under identical loading and boundary conditions, showing excellent agreement across all directions. Based on the validated model, a comprehensive parametric study is performed to quantify the influence of the unit cell’s geometric variables on stiffness and auxetic response. The results demonstrate a high degree of tunability, with particularly pronounced auxetic behavior and stiffness enhancement achievable under vertical loading through appropriate geometric design. To support the computational findings, experimental compression tests are conducted on additively manufactured polymeric and metallic lattice specimens fabricated using Digital Light Processing (DLP) and Selective Laser Melting (SLM), respectively. The combined analytical and numerical framework provides an efficient and predictive tool for the mechanics-based design and optimization of auxetic lattice metamaterials, with relevance to lightweight structural, energy-absorbing, and multifunctional engineering applications. The presented approach establishes an efficient computational mechanics framework for the predictive modeling and optimization of auxetic lattice metamaterials, offering valuable guidelines for the design of lightweight, high-performance structures in engineering applications.