Based on the fact that a search for influenza antivirals among nucleoside analogues has drawn very little attention of chemists, the present study reports the synthesis of a series of 1,2,3-triazolyl nucleoside analogues in which a pyrimidine fragment is attached to the ribofuranosyl-1,2,3-triazol-4-yl moiety by a polymethylene linker of variable length. Target compounds were prepared by the Cu alkyne-azide cycloaddition (CuAAC) reaction. Derivatives of uracil, 6-methyluracil, 3,6-dimethyluracil, thymine and quinazolin-2,4-dione with ω-alkyne substituent at the N1 (or N5) atom and azido 2,3,5-tri-O-acetyl-D-β-ribofuranoside were used as components of the CuAAC reaction. All compounds synthesized were evaluated for antiviral activity against influenza virus A/PR/8/34/(H1N1) and coxsackievirus B3. The best values of IC50 (inhibiting concentration) and SI (selectivity index) were demonstrated by the lead compound 4i in which the 1,2,3-triazolylribofuranosyl fragment is attached to the N1 atom of the quinazoline-2,4-dione moiety via a butylene linker (IC50 = 30 μM, SI = 24) and compound 8n in which the 1,2,3-triazolylribofuranosyl fragment is attached directly to the N5 atom of the 6-methyluracil moiety (IC50 = 15 μM, SI = 5). According to theoretical calculations, the antiviral activity of the 1,2,3-triazolyl nucleoside analogues 4i and 8n against H1N1 (A/PR/8/34) influenza virus can be explained by their influence on the functioning of the polymerase acidic protein (PA) of RNA-dependent RNA polymerase (RdRP).