Background: Artemisia annua, a plant with antiviral potential, has shown promise against various viral infections, yet its mechanisms of action are not fully understood. This study explores A. annua’s antiviral effects using network pharmacology and molecular docking, focusing on key active compounds and their interactions with viral protein targets, particularly within the JAK-STAT signaling pathway—a critical mediator of immune responses to viral infections. Methods: From the TCMSP database, we identified eight active compounds and 335 drug targets for A. annua, with 19 intersecting targets between A. annua compounds and viral proteins. A protein–protein interaction (PPI) network highlighted 10 key hub genes, analyzed further through Gene Ontology (GO) and KEGG pathways to understand their immune and antiviral roles. ADMET properties of the active compound Patuletin (MOL004112) were assessed, followed by 200 ns molecular dynamics simulations to examine its stability in complex with JAK2. Results: PPI analysis identified JAK2, MAPK3, MAPK1, JAK1, PTPN1, HSPA8, TYK2, RAF1, MAPT, and HMOX1 as key hub genes, with JAK2 emerging as a critical regulator of immune and antiviral pathways. ADMET analysis confirmed Patuletin’s favorable pharmacokinetic properties, and molecular dynamics simulations showed a stable Patuletin-JAK2 complex, with FEL analysis indicating minimal disruption to JAK2’s intrinsic flexibility. Conclusions: These findings highlight JAK2 as a promising target in the antiviral activity of A. annua compounds, particularly Patuletin, supporting its potential as an antiviral agent and providing a foundation for further research on A. annua’s therapeutic applications.