Saito's vanishing theorem serves as a vast generalization of Kodaira vanishing in the sense that almost all of the classical and robust vanishing theorems regarding ample line bundles can be deduced from Saito's vanishing theorem. As an application of an L^2-theoretic study of degeneration (and variation) of Hodge structures, we present a new proof of Saito's vanishing theorem going back to the original idea of Kodaira. This method gives Saito's vanishing theorem also for complex Hodge modules in the sense of Sabbah-Schnell, which does not require the mathbb{Q}-structure which is necessary for the theory of Saito's Hodge modules.