Abstract Consider an endomorphism of an algebraic variety over an algebraically closed field of characteristic zero that is injective on the complement of a proper closed subvariety. We prove that such an endomorphism is an automorphism, provided the morphism is quasi-finite. We also show that if the codimension of the subvariety is at least $2$, the endomorphism is an automorphism, provided the morphism is affine, or the variety is affine, or the variety has a normalization, which is locally factorial.