We analyze mathematical models to examine how the genetic basis of fitness affects the persistence of a population suddenly encountering a harsh environment where it would go extinct without evolution. The results are relevant for novel introductions and for an established population whose existence is threatened by a sudden change in the environment. The models span a range of genetic assumptions, including identical loci that contribute to absolute fitness, a two-locus quantitative genetic model with nonidentical loci, and a model with major and minor genes affecting a quantitative trait. We find as a general (though not universal) pattern that prospects for persistence narrow as more loci contribute to fitness, in effect because selection per locus is increasingly weakened with more loci, which can even overwhelm any initial enhancement of fitness that adding loci might provide. When loci contribute unequally to fitness, genes of small effect can significantly reduce extinction risk. Indeed, major and minor genes can interact synergistically to reduce the time needed to evolve growth. Such interactions can also increase vulnerability to extinction, depending not just on how genes interact but also on the initial genetic structure of the introduced, or newly invaded, population.