In this article we prove that given quasi-isomorphic dg vector spaces and a pre-Calabi-Yau structure on one of them there exists a pre-Calabi-Yau structure on the other as well as a pre-Calabi-Yau morphism between the two. This extends the Homotopy Transfer Theorem proved by T. Kadeishvili and a similar result of D. Petersen. Moreover, we show that a pre-Calabi-Yau morphism whose first component is an isomorphism of dg vector spaces is a pre-Calabi-Yau isomorphism, i.e. the inverse of the isomorphism of dg vector spaces extends to an inverse of the pre-Calabi-Yau morphism. We incidentally show that any pre-Calabi-Yau algebra has a minimal model. Finally, we prove that any quasi-isomorphism of pre-Calabi-Yau algebras admits a quasi-inverse.