The monolayer Janus MoSSe is different from its parent materials MoS₂ and MoSe₂. It is of great significance to study the unique mechanical properties of monolayer Janus MoSSe under uniaxial strain due to the asymmetry of its structure. Theoretical studies can provide useful support for improving the mechanical properties of monolayer Janus materials under strain. By using the first-principles method based on the density functional theory and combining the classical mechanics analysis, the mechanical properties of monolayer Janus MoSSe with broken symmetry under uniaxial tensile strain at different chiral angles are investigated. The results show that the stress-strain curves are isotropic at different chiral angles when the strain is less than 5%. When the strain exceeds 5% and the Mo—S bond and Mo—Se bond are not broken, the stress-strain curves at different chiral angles show strong anisotropic responses. The strength and toughness of monolayer Janus MoSSe are highly anisotropy- and chirality-dependent. In contrast, its in-plane stiffness remains constant at different chiral angles. By comparing the results from the first-principles method of quantum mechanics with those from the classical mechanics method, it is shown that first-principles calculations involving many-body interactions between electrons play an important role in determining the strength and toughness of this material. This is because the first-principles method can incorporate more accurately the many-body interactions between electrons. This study provides guidance for constructing and developing monolayer Janus MoSSe based nanomechanical devices.