1816, "underhand" (in reference to a style of throwing), from under + arm (n.1). First attested 1908 in dressmaking sense of "seams on the lower half of the arm-hole;" as a euphemism for armpit, it is attested from 1930s, popularized by advertisers.
underarm un·der·arm (ŭn'dər-ärm')adj. Located, placed, or used under the arm. n. The armpit.