"to feed on grass," Old English grasian, from græs "grass" (see grass). Compare Middle Dutch, Middle High German grasen, Dutch grazen, German grasen. Transitive sense from 1560s. Figurative use by 1570s. Related: Grazed; grazing.
"to touch lightly in passing," c. 1600, perhaps a transferred sense from graze (v.1) via a notion of cropping grass right down to the ground (compare German grasen "to feed on grass," used in military sense in reference to cannonballs that rebound off the ground). Related: Grazed; grazing. As a noun from 1690s, "an act of grazing."