Score-based Generative Models (SGMs) have demonstrated remarkable generalization abilities, e.g. generating unseen, but natural data. However, the greater the generalization power, the more likely the unintended generalization, and the more dangerous the abuse. Research on moderated generalization in SGMs remains limited. To fill this gap, we first examine the current 'gold standard' in Machine Unlearning (MU), i.e., re-training the model after removing the undesirable training data, and find it does not work in SGMs. Further analysis of score functions reveals that the MU 'gold standard' does not alter the original score function, which explains its ineffectiveness. Based on this insight, we propose the first Moderated Score-based Generative Model (MSGM), which introduces a novel score adjustment strategy that redirects the score function away from undesirable data during the continuous-time stochastic differential equation process. Extensive experimental results demonstrate that MSGM significantly reduces the likelihood of generating undesirable content while preserving high visual quality for normal image generation. Albeit designed for SGMs, MSGM is a general and flexible MU framework that is compatible with diverse diffusion architectures (SGM and DDPM) and training strategies (re-training and fine-tuning), and enables zero-shot transfer of the pre-trained models to downstream tasks, e.g. image inpainting and reconstruction. The code will be shared upon acceptance.