Atomically thin films of layered chromium triiodide (CrI3) have recently been regarded as suitable candidates for a wide spectrum of technologically relevant applications, mainly owing to the opportunities they offer for achieving a reversible transition between coexisting in-plane ferromagnetic and out-of-plane antiferromagnetic orders. However, no routes for inducing such a transition have been designed down to the single-layer limit. Here, we address the magnetic response of monolayer CrI3 to in-plane lattice deformations through a combination of isotropic Heisenberg spin Hamiltonians and first-principles calculations. Depending on the magnitude and orientation of the lattice strain exerted, we unveil a series of direction-dependent parallel-to-antiparallel spins crossovers, which yield the emergence of ferromagnetic, Neel antiferromagnetic, zigzag, and stripy antiferromagnetic ground states. Additionally, we identify a critical point in the magnetic phase diagram where the exchange couplings vanish and the magnetism is quenched. Our work establishes guidelines for extensively tailoring the spin interactions in the monolayer CrI3 via strain engineering and further expands the magnetically ordered phases which can be hosted in a two-dimensional crystal.