The persistence of ferromagnetic long-range order in monolayers of the van der Waals semiconductor CrI₃ opens new routes for spintronic applications based on two-dimensional quantum magnets. In the fabrication of such devices, the constituent materials inevitably experience anisotropic strain, which modifies their intrinsic electronic properties. At the same time, strain can serve as a powerful tuning parameter, driving the material to desired regimes. While several theoretical studies have investigated the effect of biaxial in-plane strain on CrI₃ numerically, experiments are widely limited to the application of hydrostatic pressure. Here, we perform high-resolution magnetostriction experiments on bulk CrI₃ samples, and ab-initio-based magnetoelastic calculations, to elucidate the role of uniaxial lattice strain on the magnetic properties. Our data show that magnetostriction in CrI₃ is unexpectedly sensitive to surface effects, which enables us to investigate the influence of in-plane and out-of-plane strain separately, in both the bulk ferromagnetic (BFM) phase emerging at T_(rm C)=61 K and the surface antiferromagnetic (SAFM) phase below T^* ≃ 50 K. In particular, we quantify the uniaxial strain dependence of the surface interlayer coupling J^(rm SAFM)_⊥ and the surface spin-flip field B^*, which drastically exceed the strain effects in the BFM phase by a factor of ∼ 30. The large magnetostrictive response allows us to study the magnetoelastic coupling in few-layer CrI₃ through experiments on bulk single crystals, without requiring exfoliation.