Six 1,4-disubstituted pyrazoles linked to a benzenesulfonamide and a benzodioxane unit have been synthesized through a copper(I)-catalyzed formal [3+2] cycloaddition (32CA) reaction of alkynes with 3-arylsydnones. The Cu-catalyzed sydnone–alkyne cycloaddition (CuSAC) procedure has been optimized to promote the formation of the pyrazole ring and to deliver in three steps the six target compounds 5a–f, fully characterized by ¹H/¹³C-NMR and mass spectrometry (EIMS). Ten solvent conditions were evaluated. The reaction proceeded most efficiently in the presence of copper(II) sulfate pentahydrate in aqueous t-butanol in the presence sodium acetate, to reach a yield of 96%. The mechanism of the Cu(I)-catalyzed reaction has been studied within the Molecular Electron Density Theory (MEDT). This rection is a domino process that consists in a Cu(I)-catalyzed formal [3+2] cycloaddition followed of an extrusion of CO₂ yielding the final pyrazole. The capacity of heterocyclic compounds 5a–f to interact with human cyclophilin A (Cyp A), which is a host cofactor for hepatitis C virus (HCV) and human immunodeficiency virus 1 (HIV-1), and with the HIV-1 protein gp120-CD4 was evaluated using molecular docking. Compounds 5a,b,d,f showed a satisfactory protein binding capacity. The physicochemical and metabolic properties of the compounds were also evaluated in silico. These predictions provide important information to guide future design in this series of potential antiviral agents.