We consider a one-loop finite N =1 supersymmetric theory in such a renormalization scheme that the first L contributions to the gauge β -function and the first (L -1 ) contributions to the anomalous dimension of the matter superfields and to the Yukawa β -function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa β -function in the first nontrivial order are valid for an arbitrary renormalization prescription respecting the above assumption. This implies that under this assumption the (L +1 ) -loop contribution to the gauge β -function and the L-loop contribution to the Yukawa β -function are always expressed in terms of the L-loop contribution to the anomalous dimension of the matter superfields. This statement generalizes the result of Grisaru, Milewski, and Zanon that for a theory finite in L loops the (L +1 ) -loop contribution to the β -function also vanishes. In particular, it gives a simple explanation why their result is valid although the NSVZ equation does not hold in an arbitrary subtraction scheme.