Deep learning-based methods have produced significant gains for hyperspectral image (HSI) classification in recent years, leading to high impact academic achievements and industrial applications. Despite the success of deep learning-based methods in HSI classification, they still lack the robustness of handling unknown object in open-set environment (OSE). Open-set classification is to deal with the problem of unknown classes that are not included in the training set, while in closed-set environment (CSE), unknown classes will not appear in the test set. The existing open-set classifiers almost entirely rely on the supervision information given by the known classes in the training set, which leads to the specialization of the learned representations into known classes, and makes it easy to classify unknown classes as known classes. To improve the robustness of HSI classification methods in OSE and meanwhile maintain the classification accuracy of known classes, a spectral-spatial latent reconstruction framework which simultaneously conducts spectral feature reconstruction, spatial feature reconstruction and pixel-wise classification in OSE is proposed. By reconstructing the spectral and spatial features of HSI, the learned feature representation is enhanced, so as to retain the spectral-spatial information useful for rejecting unknown classes and distinguishing known classes. The proposed method uses latent representations for spectral-spatial reconstruction, and achieves robust unknown detection without compromising the accuracy of known classes. Experimental results show that the performance of the proposed method outperforms the existing state-of-the-art methods in OSE.