Six novel 4-hydroxycoumarin derivatives were rationally synthesized, verified, and characterized by molecular docking using crystal HIV-1 protease. Molecular docking studies predicted antiprotease activity of (7) and (10). The most significant functional groups, responsible for the interaction with HIV-1 protease by hydrogen bonds formation are pyran oxygen, atom, lactone carbonyl oxygen and one of the hydroxyl groups. The newly synthesized compounds were biologically tested in MT-4 cells for inhibiting HIV-1 replication, exploring the protection of cells from the cytopathic effect of HIV measured by cell survival in MTT test. One derivative −7 showed 76–78% inhibition of virus infectivity with IC50 = 0.01 nM, much less than the maximal nontoxic concentration (1 mM). Antiprotease activity of 7 in two different concentrations was detected to be 25%. Nevertheless, the results of study of (7) encourage using it as a pharmacophore for further synthesis and evaluation of anti-HIV activity.