Class-Incremental Unsupervised Domain Adaptation (CI-UDA) aims to adapt a model from a labeled source domain to an unlabeled target domain, where the sets of potential target classes appearing at different time steps are disjoint and are subsets of the source classes. The key to solving this problem lies in avoiding catastrophic forgetting of knowledge about previous target classes during continuously mitigating the domain shift. Most previous works cumbersomely combine two technical components. On one hand, they need to store and utilize rehearsal target sample from previous time steps to avoid catastrophic forgetting; on the other hand, they perform alignment only between classes shared across domains at each time step. Consequently, the memory will continuously increase and the asymmetric alignment may inevitably result in knowledge forgetting. In this paper, we propose to mine and preserve domain-invariant and class-agnostic knowledge to facilitate the CI-UDA task. Specifically, via using CLIP, we extract the class-agnostic properties which we name as ''attribute''. In our framework, we learn a ''key-value'' pair to represent an attribute, where the key corresponds to the visual prototype and the value is the textual prompt. We maintain two attribute dictionaries, each corresponding to a different domain. Then we perform attribute alignment across domains to mitigate the domain shift, via encouraging visual attention consistency and prediction consistency. Through attribute modeling and cross-domain alignment, we effectively reduce catastrophic knowledge forgetting while mitigating the domain shift, in a rehearsal-free way. Experiments on three CI-UDA benchmarks demonstrate that our method outperforms previous state-of-the-art methods and effectively alleviates catastrophic forgetting. Code is available at https://github.com/RyunMi/VisTA.