The influenza virus genome features a very high mutation rate leading to the rapid selection of drug-resistant strains. Due to the emergence of drug-resistant strains, there is a need for the further development of new potent antivirals against influenza with a broad activity spectrum. Thus, the search for a novel, effective broad-spectrum antiviral agent is a top priority of medical science and healthcare systems. In this paper, derivatives based on fullerenes with broad virus inhibiting activities in vitro against a panel of influenza viruses were described. The antiviral properties of water-soluble fullerene derivatives were studied. It was demonstrated that the library of compounds based on fullerenes has cytoprotective activity. Maximum virus-inhibiting activity and minimum toxicity were found with compound 2, containing residues of salts of 2-amino-3-cyclopropylpropanoic acid (CC50 > 300 µg/mL, IC50 = 4.73 µg/mL, SI = 64). This study represents the initial stage in a study of fullerenes as anti-influenza drugs. The results of the study lead us conclude that five leading compounds (1–5) have pharmacological prospects.