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In contrast to the intensively investigated transition metal dichalcogenides like MoS2, ReS2 crystals possess a reduced in-plane symmetry, leading to anisotropic optical properties. Here, we report on the impact of strain on the Raman response of a ReS2 monolayer. Since mechanical strain can be used to shift the Raman-active phonon frequencies, we apply uniaxial tensile strain of up to 0.74% along the Re-chain direction (x axis) of the atomically thin crystal and measure the Raman response with the scattered light polarized parallel and perpendicular to the strain direction along the x axis. Complementarily, we carry out ab initio calculations to determine the phonon energies and Raman intensities under strain. We find a shift to lower energies for all phonon modes when tensile strain is applied to the monolayer. The determined gauge factors/Grüneisen parameters are in good agreement between experiment and theory. Our study demonstrates that the optomechanical properties of ReS2 can be tuned by external straining, which is of importance for potential future strain-sensor applications, e.g., in the biomedical sector.