The c-Myc protein, a key regulator of cell proliferation, growth, and apoptosis in B-cell lymphocytes, is frequently dysregulated in Burkitt’s lymphoma. Zingiberaceae plants—galangal (Alpinia galanga), black turmeric (Curcuma aeroginosa), black ginger (Kaempferia parviflora), phlai lueang (Zingiber montanum), and phlai dum (Zingiber ottensii)—are traditionally used as herbal remedies and may serve as natural anti-lymphoma agents. In this study, extracts from these five plants were screened for cytotoxicity against Raji and Daudi lymphoma cell lines and compared with their effects on normal peripheral blood mononuclear cells (PBMCs). Galangal extract exhibited the strongest cytotoxic effects on lymphoma cells. Its major bioactive compounds, galangin and 1′-acetoxychavicol acetate (ACA), showed selective cytotoxicity, with ACA being more potent. ACA significantly suppressed both c-Myc and phosphorylated c-Myc (p-c-Myc) protein levels and induced dose-dependent apoptosis in lymphoma cells. Cell cycle analysis revealed arrest at specific phases, supporting its anti-proliferative action. Furthermore, network pharmacology and pathway enrichment analyses implicated ACA in the modulation of oncogenic PI3K-Akt and MAPK pathways. These findings highlight ACA as a promising plant-derived therapeutic candidate for lymphoma, acting through c-Myc suppression, cell cycle arrest, and apoptosis induction.