Image denoising and classification are typically conducted separately and sequentially according to their respective objectives. In such a setup, where the two tasks are decoupled, the denoising operation does not optimally serve the classification task and sometimes even deteriorates it. We introduce here a unified deep learning framework for joint denoising and classification of high-dimensional images, and we particularly apply it in the framework of hyperspectral imaging. Earlier works on joint image denoising and classification are very scarce, and to the best of our knowledge, no deep learning models were proposed or studied yet for this type of multitask image processing. A key component in our joint learning model is a compound loss function, designed in such a way that the denoising and classification operations benefit each other iteratively during the learning process. Hyperspectral images (HSIs) are particularly challenging for both denoising and classification due to their high dimensionality and varying noise statistics across the bands. We argue that a well-designed end-to-end deep learning framework for joint denoising and classification is superior to current deep learning approaches for processing HSI data, and we substantiate this by results on real HSI images in remote sensing. We experimentally show that the proposed joint learning framework substantially improves the classification performance compared to the common deep learning approaches in HSI processing, and as a by-product, the denoising results are enhanced as well, especially in terms of the semantic content, benefiting from the classification.