Early detection of grapevine viral diseases is critical for early interventions in order to prevent the disease from spreading to the entire vineyard. Hyperspectral remote sensing can potentially detect and quantify viral diseases in a nondestructive manner. This study utilized hyperspectral imagery at the plant level to identify and classify grapevines inoculated with the newly discovered DNA virus grapevine vein-clearing virus (GVCV) at the early asymptomatic stages. An experiment was set up at a test site at South Farm Research Center, Columbia, MO, USA (38.92 N, −92.28 W), with two grapevine groups, namely healthy and GVCV-infected, while other conditions were controlled. Images of each vine were captured by a SPECIM IQ 400–1000 nm hyperspectral sensor (Oulu, Finland). Hyperspectral images were calibrated and preprocessed to retain only grapevine pixels. A statistical approach was employed to discriminate two reflectance spectra patterns between healthy and GVCV vines. Disease-centric vegetation indices (VIs) were established and explored in terms of their importance to the classification power. Pixel-wise (spectral features) classification was performed in parallel with image-wise (joint spatial–spectral features) classification within a framework involving deep learning architectures and traditional machine learning. The results showed that: (1) the discriminative wavelength regions included the 900–940 nm range in the near-infrared (NIR) region in vines 30 days after sowing (DAS) and the entire visual (VIS) region of 400–700 nm in vines 90 DAS; (2) the normalized pheophytization index (NPQI), fluorescence ratio index 1 (FRI1), plant senescence reflectance index (PSRI), anthocyanin index (AntGitelson), and water stress and canopy temperature (WSCT) measures were the most discriminative indices; (3) the support vector machine (SVM) was effective in VI-wise classification with smaller feature spaces, while the RF classifier performed better in pixel-wise and image-wise classification with larger feature spaces; and (4) the automated 3D convolutional neural network (3D-CNN) feature extractor provided promising results over the 2D convolutional neural network (2D-CNN) in learning features from hyperspectral data cubes with a limited number of samples.