Oral cancer is a severe health problem that accounts for an alarmingly high number of fatalities worldwide. Withania somnifera (L.) Dunal has been extensively studied against various tumor cell lines from different body organs, rarely from the oral cavity. We thus investigated the cytotoxicity of W. somnifera fruits (W-F) and roots (W-R) hydromethanolic extracts and their chromatographic fractions against oral squamous cell carcinoma (OSCC) cell lines [Ca9-22 (derived from gingiva), HSC-2, HSC-3, and HSC-4 (derived from tongue)] and three normal oral mesenchymal cells [human gingival fibroblast (HGF), human periodontal ligament fibroblast (HPLF), and human pulp cells (HPC)] in comparison to standard drugs. The root polar ethyl acetate (W-R EtOAc) and butanol (W-R BuOH) fractions exhibited the strongest cytotoxicity against the Ca9-22 cell line (CC50 = 51.8 and 40.1 μg/mL, respectively), which is relatively the same effect as 5-FU at CC50 = 69.4 μM and melphalan at CC50 = 36.3 μM on the same cancer cell line. Flow cytometric analysis revealed changes in morphology as well as in the cell cycle profile of the W-R EtOAc and W-R BuOH-treated oral cancer Ca9-22 cells compared to the untreated control. The W-R EtOAc (125 μg/mL) exerted morphological changes and induced subG1 accumulation, suggesting apoptotic cell death. A UHPLC MS/MS analysis of the extract enabled the identification of 26 compounds, mainly alkaloids, withanolides, withanosides, and flavonoids. Pharmacophore-based inverse virtual screening proposed that BRD3 and CDK2 are the cancer-relevant targets for the annotated withanolides D (18) and O (12), and the flavonoid kaempferol (11). Molecular modeling studies highlighted the BRD3 and CDK2 as the most probable oncogenic targets of anticancer activity of these molecules. These findings highlight W. somnifera’s potential as an affordable source of therapeutic agents for a range of oral malignancies.