As a newly synthesised two-dimensional (2D) carbon material, mechanical properties of the monolayer fullerene network or, namely, quasi-hexagonal-phase fullerene (qHP C60) [Hou et al., Nature, 606, 507-510 (2022)] are almost unexplored. In this work, first-principles calculations were conducted to reveal the elastic properties and fracture behaviours of monolayer qHP C60, both of which are strongly anisotropic. Specifically, the Young’s modulus (166 GPa) and the Poisson’s ratio (0.12) in the minimum direction of monolayer qHP C60 are both smaller than any other synthesized 2D carbon crystal materials. In addition, the qHP C60 monolayer stretched along the direction of the [2 + 2] cycloaddition bond is found to possess a larger tensile strength (fracture strain) and exhibit a more complicated crack pattern. The robustness of monolayer qHP C60 under mechanical loading and the sensitivity of its band gap to loading suggest that the strain engineering is an effective method to modulate the electronic properties of monolayer qHP C60, which is found to be especially more efficient when the loading is perpendicular to the [2 + 2] cycloaddition bond. These findings suggest that monolayer qHP C60 could be a potential candidate in developing flexible 2D electronic devices with strain-tunable performance.