We report an unprecedented highly enantioselective desymmetric Cu(I)-catalyzed 1,3-dipolar cycloaddition of diazides with terminal alkynes and 1-iodoalkynes, affording tertiary alcohols bearing a 1,2,3-triazole moiety in high yield and excellent ee value. PYBOX ligands with a C4 shielding group once again show the promising ability to achieve higher enantioselectivity.