Recent experimental evidence demonstrates that shifts in mutational biases, for example increases in transversion frequency, can change the distribution of fitness effects of mutations (DFE). In particular, reducing or reversing a prevailing bias can increase the probability that a de novo mutation is beneficial. It has also been shown that mutator bacteria are more likely to emerge if the beneficial mutations they generate have a larger effect-size than observed in the wildtype. Here, we connect these two results, demonstrating that mutator strains that reduce or reverse a prevailing bias have a positively-shifted DFE, which in turn can dramatically increase their emergence probability. Since changes in mutation rate and bias are often coupled through the gain and loss of DNA repair enzymes, our results predict that the invasion of mutator strains will be facilitated by shifts in mutation bias that offer improved access to previously under-sampled beneficial mutations.