The introduction of the copper-catalyzed azide− alkyne coupling (CuAAC) to 1,3-dipolar cycloadditions was pivotal to their popularization in synthetic chemistry and to their application to multiple other domains of science. The reaction rate enhancement observed when coinage metal acetylide intermediates are involved in the cyclization process greatly expanded the structural and conditional range in which (3 + 2) cycloadditions may take place with terminal alkynes. Herein, we report that comparable rate enhancements, in nature and level, are induced by copper and silver catalysts in the intramolecular (3 + 2) cycloaddition of terminal alkynes with "neutral" three-atom components (TACs), specifically alkynyl sulfides. Through careful observations amidst reaction optimization, experimental, and DFT mechanistic studies, a pathway involving a proton-coupled cyclometallation key step is proposed. The sets of catalytic conditions that have been developed allow us to overcome several scope limitations previously presented by the thermally promoted (3 + 2) cycloaddition of "neutral" TACs, thus expanding their synthetic and applicative potential.