Positively curved manifolds of cohomogeneity one have been classified by Verdiani in even dimensions and by Grove, Wilking and Ziller in odd dimensions except for two families in dimension seven. We extended these classification results with the curvature condition relaxed to quasipositive sectional curvature. The proof is based on the work of Grove, Wilking and Ziller. Three main tools that we extended to quasipositively curved cohomogeneity one manifolds are Wilking's Chain Theorem, the classification of positively curved fixed point homogeneous manifolds by Grove and Searle and the well known Rank Lemma.