Although Lorentz invariance forbids the presence of a term that tilts the energy-momentum relation in the Weyl Hamiltonian, a tilted dispersion is not forbidden and, in fact, generic for condensed matter realizations of Weyl semimetals. We here investigate the combined effect of such a tilted Weyl dispersion and the presence of potential disorder. In particular, we address the influence of a tilt on the disorder-induced phase transition between a quasiballistic phase at weak disorder, in which the disorder is an irrelevant perturbation, and a diffusive phase at strong disorder. Our main result is that the presence of a tilt leads to a reduction of the critical disorder strength for this transition or, equivalently, that increasing the tilt at fixed disorder strength drives the system through the phase transition to the diffusive strong-disorder phase. Notably this obscures the tilt-induced Lifshitz transition to an overtilted type II Weyl phase at any finite disorder strength. Our results are supported by analytical calculations using the self-consistent Born approximation and numerical calculations of the density of states and of transport properties.