Vector-borne flaviviruses (VBFs) affect human health worldwide, but no approved drugs are available specifically to treat VBF-associated infections. Here, we performed in silico screening of a library of U.S. Food and Drug Administration-approved antiviral drugs for their interaction with Zika virus proteins. Twelve hit drugs were identified by the docking experiments and tested in cell-based antiviral assay systems. Efavirenz, tipranavir, and dasabuvir at micromolar concentrations were identified to inhibit all VBFs tested; i.e., two representatives of mosquito-borne flaviviruses (Zika and West Nile viruses) and one representative of flaviviruses transmitted by ticks (tick-borne encephalitis virus). The results warrant further research into these drugs, either individually or in combination, as possible pan-flavivirus inhibitors.