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The Triple Periodic Minimal Surface (TPMS) has emerged as a new approach for producing open cell porous scaffolds for biomedical applications. However, unlike the traditional scaffolds, the TPMS scaffolds always exhibit anisotropic elastic behaviors and consequently the simple mechanical testing is not capable to provide a full characterization of their mechanical behaviors. The aim of the present study was to analyze the anisotropic elastic behaviors of TPMS-based scaffolds using the numerical homogenization method and the analytical analysis approach. Five widely-used TPMS scaffold topologies (Diamond, Gyroid, Fischer-Koch S, Schwarz P and F-RD) were investigated. The independent elastic constants were determined from the analytical analysis and then, the values for these independent constants were determined using the finite element (FE) analysis of the scaffold unit cell models combined with the periodic boundary condition. The analytical analysis revealed that the Diamond, Gyroid and Fischer-Koch S topologies are threefold rotational symmetric and consequently have seven independent elastic constants. The Schwarz P and F-RD topologies are cubic symmetric and have three independent elastic constants. The FE analysis showed that the Diamond, Gyroid and Fischer-Koch S-based scaffolds have only three non-zero independent elastic constants, implying the cubic symmetric property of these scaffolds. All the independent elastic constants decreased quadratically with the increase of scaffold porosity. The absolute difference between the Zener anisotropic factor and 1.0 increased the most for the Gyroid-based scaffold, while the value for the Fischer-Koch S-based scaffold increased the least. The present study provided some novel insights into the anisotropic elastic behavior of TPMS-based scaffolds and can facilitate the selection and design of scaffold in biomedicine and relevant fields.