Cross-scene hyperspectral image (HSI) classification stands as a fundamental research topic in remote sensing, with extensive applications spanning various fields. Owing to the inclusion of unknown categories in the target domain and the existence of domain shift across different scenes, open-set domain adaptation techniques are commonly employed to address cross-scene HSI classification. However, existing open-set cross-scene HSI classification methods still face two critical challenges: (1) domain shift issues arising from the direct alignment of mixed spectral-spatial features; (2) high computational costs caused by two-stage training strategies. To address these issues, this paper proposes a single-stage open-set domain adaptation method with decoupled alignment (SoDa$^2$) for cross-scene HSI classification. A contribution-aware dual-modality feature extraction is customized to disentangle the characteristics from spectral sequence signals and spatial details, selectively and adaptively enhancing discriminative features. The decoupled alignment module minimizes the Maximum Mean Discrepancy to independently reduce the spectral discrepancy and the spatial discrepancy between the source and target domains, extracting more fine-grained domain-invariant features. A cost-effective single-stage dual-branch framework is designed to learn MMD-constrainted aligned features and constraint-free intrinsic features for adaptive distinction between known and unknown classes. This framework employs a Gaussian Mixture Model to model the squared cosine similarity distribution between the two feature types, enabling open-set recognition without prior knowledge of unknown classes. Extensive experiments on three groups of HSI datasets demonstrate that SoDa$^2$ outperforms state-of-the-art methods, achieving superior classification accuracy and model transferability for open-set cross-scene tasks.