Cladiella-derived natural products have shown promising anticancer properties against many human cancer cell lines. In the present investigation, we found that an ethyl acetate extract of Cladiella pachyclados (CE) collected from the Red Sea could inhibit the human breast cancer (BC) cells (MCF and MDA-MB-231) in vitro (IC50 24.32 ± 1.1 and 9.55 ± 0.19 µg/mL, respectively). The subsequent incorporation of the Cladiella extract into the green synthesis of silver nanoparticles (AgNPs) resulted in significantly more activity against both cancer cell lines (IC50 5.62 ± 0.89 and 1.72 ± 0.36, respectively); the efficacy was comparable to that of doxorubicin with much-enhanced selectivity. To explore the mode of action of this extract, various in silico and network-pharmacology-based analyses were performed in the light of the LC-HRESIMS-identified compounds in the CE extract. Firstly, using two independent machine-learning-based prediction software platforms, most of the identified compounds in CE were predicted to inhibit both MCF7 and MDA-MB-231. Moreover, they were predicted to have low toxicity towards normal cell lines. Secondly, approximately 242 BC-related molecular targets were collected from various databases and used to construct a protein–protein interaction (PPI) network, which revealed the most important molecular targets and signaling pathways in the pathogenesis of BC. All the identified compounds in the extract were then subjected to inverse docking against all proteins hosted in the Protein Data bank (PDB) to discover the BC-related proteins that these compounds can target. Approximately, 10.74% of the collected BC-related proteins were potential targets for 70% of the compounds identified in CE. Further validation of the docking results using molecular dynamic simulations (MDS) and binding free energy calculations revealed that only 2.47% of the collected BC-related proteins could be targeted by 30% of the CE-derived compounds. According to docking and MDS experiments, protein-pathway and compound-protein interaction networks were constructed to determine the signaling pathways that the CE compounds could influence. This paper highlights the potential of marine natural products as effective anticancer agents and reports the discovery of novel anti-breast cancer AgNPs.