late 14c., "device put over an animal's mouth to stop it from biting, eating, or rooting," from Old French musel "muzzle," also "snout, nose" (12c., Modern French museau), from muse "muzzle," from Gallo-Roman *musa "snout" (source also of Provençal mus, Old Spanish mus, Italian muso), of unknown origin, possibly related to Latin morsus "bite" (but OED finds "serious difficulties" with this). Meaning "projecting part of the head of an animal" is from early 15c. in English; sense of "open end of a firearm" first recorded 1560s.

"to put a muzzle on," early 15c., from muzzle, n.. Figurative use from 1610s. Related: Muzzled; muzzling.