Unsupervised domain adaptation is critical to many real-world applications where label information is unavailable in the target domain. In general, without further assumptions, the joint distribution of the features and the label is not identifiable in the target domain. To address this issue, we rely on the property of minimal changes of causal mechanisms across domains to minimize unnecessary influences of distribution shifts. To encode this property, we first formulate the data-generating process using a latent variable model with two partitioned latent subspaces: invariant components whose distributions stay the same across domains and sparse changing components that vary across domains. We further constrain the domain shift to have a restrictive influence on the changing components. Under mild conditions, we show that the latent variables are partially identifiable, from which it follows that the joint distribution of data and labels in the target domain is also identifiable. Given the theoretical insights, we propose a practical domain adaptation framework called iMSDA. Extensive experimental results reveal that iMSDA outperforms state-of-the-art domain adaptation algorithms on benchmark datasets, demonstrating the effectiveness of our framework.