Cervical cancer is the fourth most diagnosed cancer and the fourth leading cause of cancer death in women globally. Its onset and progression have been attributed to high-risk human papillomavirus (HPV) types, especially 16 and 18, while the Epstein–Barr virus (EBV) is believed to also significantly contribute to cervical cancer growth. The E6 protein associated with high-risk HPV strains, such as HPV16 and HPV18, is known for its role in promoting cervical cancer and other anogenital cancers. E6 proteins contribute to the malignant transformation of infected cells by targeting and degrading tumor suppressor proteins, especially p53. On the other hand, EBV nuclear antigen 1 (EBNA1) plays a crucial role in the maintenance and replication of the EBV genome in infected cells. EBNA1 is believed to increase HPV E6 and E7 levels, as well as c-MYC, and BIRC5 cellular genes in the HeLa cell line, implying that HPV/EBV co-infection accelerates cervical cancer onset and growth. Thus, the E6 and EBNA1 antigens of HPV and EBV, respectively, are attractive targets for cervical cancer immunotherapy. This study, therefore, virtually screened for potential drug candidates with good binding affinity to all three oncoviral proteins, HPV16 E6, HPV18 E6, and EBNA1. The compounds were further subjected to ADMET profiling, biological activity predictions, molecular dynamics (MD) simulations, and molecular mechanics Poisson–Boltzmann surface area (MM/PBSA) calculations. A total of six compounds comprising ZINC000013380012, ZINC000070454124, ZINC000014588133, ZINC000085568136, ZINC000095909247, and ZINC000085597263 demonstrated very strong affinity (≤−60 kJ/mol) to the three oncoviral proteins (EBNA1, HPV16 E6, and HPV18 E6) after being subjected to docking, MD, and MM/PBSA. These compounds demonstrated relatively stronger binding than the controls used, inhibitors of EBNA1 (VK-1727) and HPV E6 (baicalein and gossypetin). Biological activity predictions also corroborated their antineoplastic, p53-enhancing, Pin1 inhibitory, and JAK2 inhibitory activities. Further experimental testing is required to validate the ability of the shortlisted compounds to silence the insidious effects of HPV E6 and EBNA1 proteins in cervical cancers.