1560s, "to handle, train, or direct" (a horse), from the now-obsolete noun manage "the handling or training of a horse; horsemanship" (see manege, which is a modern revival of it), from Old French manège "horsemanship," from Italian maneggio, from maneggiare "to handle, touch," especially "to control a horse," which ultimately from Latin noun manus "hand" (from PIE root *man- (2) "hand").
Extended sense of "control or direct by administrative ability" any sort of business is by 1570s; meaning "to wield (a tool or object) by hand" is from 1580s. Meaning "effect by effort" (hence "succeed in accomplishing") is by 1732. Intransitive sense of "get by, carry on affairs" is suggested by 1650s, in frequent use from mid-19c. Related: Managed; managing. Managed economy was used by 1933.
Manage literally implies handling, and hence primarily belongs to smaller concerns, on which one may at all times keep his hand: as, to manage a house; to manage a theater. Its essential idea is that of constant attention to details: as, only a combination of great abilities with a genius for industry can manage the affairs of an empire. [Century Dictionary]