Malaria continues to impose a substantial burden on human health. We have previously proposed that biological approaches to control the mosquito vector of disease could be developed using homing endonuclease genes (HEGs), a class of selfish or parasitic gene that exists naturally in many microbes. Recent lab studies have demonstrated that HEGs can function in mosquitoes. We constructed and analyzed a model of mosquito population genetics and malaria epidemiology to determine how well HEGs need to function in order to have a significant effect on the burden of disease. Our model, combined with currently available data, indicates that populations of Anopheles gambiae could be eliminated by releasing 2-3 HEGs targeting female fertility genes, or a driving-Y chromosome that is transmitted to 75-96% of progeny. Combinations of fertility-targeting HEGs and Y drive may also be effective. It is possible to eliminate the disease without eliminating the vector, but the parameter space producing this outcome appears to be small. HEGs causing a quantitative reduction in adult survival can be more effective than those targeting female fertility, but the selection coefficients that need to be imposed are still large, unless many HEGs are to be released. Simulations show that HEG-based strategies can be effective over socially relevant time frames. Important limiting assumptions of the models are that there is only a single vector species, and we model a homogeneous population, not a landscape. Nevertheless, we conclude that HEG-based approaches could have a transformational effect on malaria control efforts