Background: Dengue virus (DENV) is the fatal pathogenic arthropod-borne virus (arboviruses) that belongs to the Flaviviridae family, which transmits to humans through mosquito bites from infected Aedes aegypti and Aedes albopictus mosquitoes or maternal-fetal transmission. Despite antigenic differences, the four serotypes of DENV (DENV-1 to DENV-4) share 65–78% of their genome. Non-structural (NS) proteins amongst serotypes show analogous functions. Among NS proteins, NS3 and NS5 are frequently used as targets for antiviral drugs due to their multifunctional roles. Methods: To identify potential inhibitors of DENV, we created a phytochemical library of 898 compounds derived from 17 medicinal plants recognized for their medicinal and antiviral properties. The phytochemicals library has been docked against the target proteins. Phytochemicals with a docking score greater than −8.0 kcal/mol were selected for further evaluation using a machine learning approach. Further, molecular dynamics (MD) simulations were conducted to evaluate the root mean square deviation, root mean square fluctuation, solvent-accessible surface area, radius of gyration, and hydrogen bond count of the compounds. Results: From the docking results, Silibinin, Rubiadin, and Ellagic acid showed binding affinities of −8.5, −8.3, and −8.2 kcal/mol, respectively, for NS3, and NSC 640467, Bisandrographolide A, and Andrographidin A showed binding affinities of −9.3, −10.1, and −9.3 kcal/mol, respectively, for NS5 target proteins. These compounds exhibited strong interactions with target proteins. MD simulation results confirmed the stable formation of protein–ligand complexes. Further, absorption, distribution, metabolism, excretion, and toxicity (ADMET) and bioactivity predictions confirmed their pharmacological safety. Conclusions: Despite global public health concerns, DENV still lacks specific drug treatments. Our identified new drug candidates might help for developing effective antiviral inhibitors against the DENV. However, further confirmation is needed through in vivo and in vitro research.