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Existing deep learning-based change detection methods try to elaborately design complicated neural networks with powerful feature representations, but ignore the universal domain shift induced by time-varying land cover changes, including luminance fluctuations and season changes between pre-event and post-event images, thereby producing sub-optimal results. In this paper, we propose an end-to-end Supervised Domain Adaptation framework for cross-domain Change Detection, namely SDACD, to effectively alleviate the domain shift between bi-temporal images for better change predictions. Specifically, our SDACD presents collaborative adaptations from both image and feature perspectives with supervised learning. Image adaptation exploits generative adversarial learning with cycle-consistency constraints to perform cross-domain style transformation, effectively narrowing the domain gap in a two-side generation fashion. As to feature adaptation, we extract domain-invariant features to align different feature distributions in the feature space, which could further reduce the domain gap of cross-domain images. To further improve the performance, we combine three types of bi-temporal images for the final change prediction, including the initial input bi-temporal images and two generated bi-temporal images from the pre-event and post-event domains. Extensive experiments and analyses on two benchmarks demonstrate the effectiveness and universality of our proposed framework. Notably, our framework pushes several representative baseline models up to new State-Of-The-Art records, achieving 97.34% and 92.36% on the CDD and WHU building datasets, respectively. The source code and models are publicly available at https://github.com/Perfect-You/SDACD.