1590s, "a defeat (of an army, etc.) followed by disorderly retreat," from French route "disorderly flight of troops," literally "a breaking off, rupture," from Vulgar Latin *rupta "a dispersed group," literally "a broken group," from noun use of Latin rupta, fem. past participle of rumpere "to break" (see rupture (n.)).
The archaic English noun rout "group of persons, assemblage," is the same word, from Anglo-French rute, Old French route "host, troop, crowd," from Vulgar Latin rupta "a dispersed group," here with sense of "a division, a detachment." It came to English meaning "group of soldiers" (early 13c.), also "gang of outlaws or rioters, mob" (c. 1300) before the more general sense developed 14c.: "large social assemblage, a general gathering of guests for entertainment." But it also kept its sense of "disorderly or confused mass of persons, the rabble," and was a legal term in this meaning. A rout-cake (1807) was one baked for use at a reception.
"drive (a body of troops) into disordered flight by defeat," c. 1600, from rout (n.). Hence "defeat or repulse thoroughly." Related: Routed; routing.