One of the consequences of the cubic case of Vinogradov's mean value theorem is that it made it possible to obtain close-to-optimal mean values for exponential sums related to certain incompleteVinogradov systems as well. Using the new mean value for exponential sums involving one cubic and one quadratic equation, we employ Brüdern's and Wooley's new complification method in order to establish the Hasse principle for systems of $r_3$ cubic and $r_2$ quadratic diagonal forms, when the number of cubic equations is at least twice the number of quadratic forms. In particular, we achieve essentially square root cancellation for systems of one quadratic and arbitrarily many cubic equations.