A straightforward method for the synthesis of 5-substituted tetrazolo[1,5-a]pyrido[2,3-e]pyrimidines from 2,4-diazidopyrido[3,2-d]pyrimidine in SNAr reactions with N-, O-, and S- nucleophiles has been developed. The various N- and S-substituted products were obtained with yields from 47% to 98%, but the substitution with O-nucleophiles gave lower yields (20–32%). Furthermore, the fused tetrazolo[1,5-a]pyrimidine derivatives can be regarded as 2-azidopyrimidines and functionalized in copper(I)-catalyzed azide-alkyne dipolar cycloaddition (CuAAC) and Staudinger reactions due to the presence of a sufficient concentration of the reactive azide tautomer in solution. In total, seven products were fully characterized by their single crystal X-ray studies, while five of them were representatives of the tetrazolo[1,5-a]pyrido[2,3-e]pyrimidine heterocyclic system. Equilibrium constants and thermodynamic values were determined using variable temperature 1H NMR and are in agreement of favoring the tetrazole tautomeric form (ΔG298 = −3.33 to −7.52 (kJ/mol), ΔH = −19.92 to −48.02 (kJ/mol) and ΔS = −43.74 to −143.27 (J/mol·K)). The key starting material 2,4-diazidopyrido[3,2-d]pyrimidine presents a high degree of tautomerization in different solvents.