(intransitive) "go away," 1958, chiefly British; (transitive) "annoy," 1968, chiefly U.S.; from piss
(v.) + off
. Pissed off
"angry, fed up" is attested by 1946 (Partridge says 1937); said to have been used in the military in World War II; in common use from 1970s.