1944, from a specialized use in psychology of German Angst "neurotic fear, anxiety, guilt, remorse," from Old High German angust, from Proto-Germanic *angustu-(source also of Old Frisian ongost, Old High German angust, Middle Dutch ancst "fear," also Old English enge, Old Saxon engi, Gothic aggwus "narrow"), from PIE *anghosti-, suffixed form of root *angh- "tight, painfully constricted, painful." Compare anger.
George Eliot used it (in German) in 1849, and it was popularized in English early 20c. by translation of Freud's work, but as a foreign word until 1940s. Old English had a cognate word, angsumnes "anxiety," but it died out.