The piezomagnetic effect (PZM), a linear coupling between magnetization and mechanical stress, provides a direct pathway for strain-controlled magnetic functionality. However, experimental realizations remain largely confined to a limited class of Mn-, Fe-, Co-, and Cu-based antiferromagnets, and its microscopic design principles are not fully established. Here, we report a systematic investigation of intrinsic PZM in the quasi-one-dimensional antiferromagnet NiTe2O5. Temperature- and field-dependent magnetization measurements under controlled uniaxial stress reveal a robust linear stress-induced magnetization below TN ≈ 30 K, with the induced magnetization remaining essentially independent of applied magnetic field. Magnetic symmetry analysis of the centrosymmetric magnetic space group (Pnma, k = (0, 0, 0)) restricts the linear PZM response to shear-type tensor components, and first-principles calculations identify U25 as the dominant coupling channel. Neutron diffraction reveals strong easy-axis anisotropy along the b axis and intrinsic canting in the ac plane, while temperature-dependent Raman spectroscopy provides direct evidence of pronounced spin-phonon coupling below TN. Together, these results support a mechanism in which ac shear perturbs local spin-chirality, converting intrinsic canting into a finite magnetization along the b axis. The observed piezomagnetic coefficient is comparable to several representative PZM systems, identifying NiTe2O5 as a rare Ni-based piezomagnetic material in which structural distortion enhances spin-orbit-mediated anisotropic exchange, challenging the conventional expectation of weak strain-spin coupling in nominally orbitally quenched 3d8 systems. These results establish low-dimensional Ni-based antiferromagnets as promising candidates for symmetry-driven strain-spin engineering and broaden the material landscape of intrinsic PZM.