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This study presents a comprehensive first-principles investigation of the pressure-dependent phase transitions and elastic properties of GeO2. Using density functional theory, complete sets of single-crystal elastic constants were calculated at 0 K for all structurally stable phases over a wide pressure range (0 -120 GPa). Strain analysis identifies the rutile-to-CaCl2 type transition at a critical pressure of 14.57 GPa (Ghosh et al., 2025). Moreover, this transition is a second-order phase transition. Within the framework of classical Landau theory, this transition is described by symmetry-adapted strain order parameters. We have also shown the evolution of elastic moduli with pressure using Landau coefficients obtained from the parent tetragonal phase (rutile). The results show elastic softening as the critical pressure is approached, manifested by clear anomalies in the bulk modulus and compressional wave velocity (Vp), both of which exhibit a distinct minimum near the transition pressure. Following this analysis, we have also computed the elastic constants for the α-PbO2 and pyrite-type phases of GeO2. Elastic anisotropy analysis reveals a strong mechanical instability across the tetragonal-to-orthorhombic transition, driven primarily by a rapid reduction in shear wave velocity. These results provide a unified, elastic, and symmetry-based interpretation of pressure-induced phase transitions in GeO2, with implications for understanding the mechanical stability and seismic properties of rutile-type oxides under extreme conditions.