Marine organisms and their metabolites are a diverse source of scaffolds for potential pharmacological molecular probes and, less frequently, for pharmaceutical lead compounds. In this study, 157 synthetic analogues of marine sponge-derived alkaloids clathrodin and oroidin were screened against replicon models of two RNA viruses, hepatitis C (HCV) and Chikungunya virus (CHIKV) as part of a larger screening project. Four compounds were found to selectively inhibit the HCV replicon (IC50 1.6–4.6 μm). These belong to the 4,5,6,7-tetrahydrobenzo[1,2-d]thiazole class of compounds originally designed to target the ATP-binding site of bacterial DNA gyrase. The ATP-binding site of this bacterial protein has high structural similarity to the ATP-binding site of heat-shock protein 90 (Hsp90), a host cell chaperone universally required for viral replication, which led us to examine inhibition of Hsp90 as the compounds’ potential mechanism of action. Binding of the four hit compounds to Hsp90 was evaluated through microscale thermophoresis and molecular modeling, which supported our hypothesis of interaction with Hsp90 (Kd 18–79 μm) as basis for the compounds’ antiviral activity. The presented novel structural class of small molecules that target the Hsp90 ATP-binding site has excellent potential for further antiviral drug development because of the compounds’ low toxicity and synthetic accessibility.