The critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non-Markovian biased random walks with reactions. From the Hamilton-Jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation. If the reaction rate is larger than the jump frequency, invasion always succeeds, even in the case of maximal bias. Numerical simulations support our analytical predictions.