Old English muþ "mouth, opening, door, gate," from Proto-Germanic *muntha- (source also of Old Saxon, Old Frisian muth, Old Norse munnr, Danish mund, Middle Dutch mont, Dutch mond, Old High German mund, German Mund, Gothic munþs "mouth"), with characteristic loss of nasal consonant in Old English (compare tooth, goose), probably an IE word, but the exact etymology is disputed. Perhaps from the source of Latin mentum "chin" (from PIE root *men- (2) "to project," on the notion of "projecting body part"), presuming a semantic shift from "chin" to "mouth."
In the sense of "outfall of a river" it is attested from late Old English; as the opening of anything with capacity (a bottle, cave, etc.) it is recorded from c. 1200. Mouth-organ attested from 1660s.