In this paper, we have presented the structural, elastic, mechanical, and electronic properties of the transition metal chalcogenide perovskite SnZrS3 under different pressures by using first-principles method. Our calculated lattice parameters at ambient pressure are in good agreement with the experimental and previous theoretical results. The elastic constants were evaluated numerically for orthorhombic SnZrS3 using the strain-stress approach. Orthorhombic SnZrS3 shows a strong anisotropic behavior of the elastic and structural properties. According to the calculations of the electronic properties, we find the states near the valence band top are derived from S 3p, Zr 4d, Sn 5p, and Sn 5s orbitals, and the lowest conduction band is composed of Zr 4d, S 3p, and Sn 5p orbitals. As the pressure increases, the conduction and valence band shift to lower and higher energies, respectively. These results indicated that lattice constants and band gap decrease with the increase of pressure.