Invasive rodents, including house mice, are a major cause of environmental damage and biodiversity loss, particularly in island ecosystems. Eradication can be achieved through the distribution of rodenticide, but this approach is expensive to apply at scale, can have negative impacts (e.g. on non-target species, or through contamination), has animal ethics concerns, and has restrictions on where it can be used. Gene drives, which exhibit biased inheritance, have been proposed as a next generation strategy to control invasive alien pests and disease vectors. However, synthetic gene drives including CRISPR homing drives have proven to be technically challenging to develop in mice. The t haplotype is a naturally-occurring segregation distortion locus with highly biased transmission from heterozygous males. Here we propose a novel gene drive strategy for population suppression, tCRISPR, that leverages t haplotype bias and an embedded SpCas9/gRNA transgene to spread inactivating mutations in a haplosufficient female fertility gene. Using spatially explicit individual-based in silico modelling, we show that polyandry, sperm competition, dispersal, and transmission bias are critical factors for tCRISPR-mediated population suppression. Modelling of realistic parameter values indicates that tCRISPR can eradicate an island population of 200,000 mice while the unmodified t haplotype fails under the same conditions. We also demonstrate feasibility of this approach by engineering tCRISPR mice in a safe split drive format. tCRISPR mice exhibit high transmission of the modified t haplotype, and efficient generation and transmission of inactivating mutations in a recessive female fertility gene, crucially, at levels for which the modelling predicts that population eradication can occur. This is the first example of a feasible gene drive system for invasive alien rodent population control.