Origins
*Save* arrived in English by the most travelled of routes: Latin → Old French → Middle English. Its immediate parent is Old French *sauver* 'to keep safe, deliver, redeem', which English took on around 1200 as Middle English *saven*. *Sauver* in turn descends from Late Latin *salvāre* 'to make safe', a verb built directly on the classical Latin adjective *salvus* — 'sound, uncorrupted, healthy, safe'. *Salvus* sat at the centre of a small constellation of Latin words: the noun *salus* 'health, well-being', the greeting *salve!* 'be well', and the adjective *salutaris* 'conducive to health', which gives English *salutary*.
Latin *salvus* itself goes back to a Proto-Indo-European root reconstructed as **solh₂-*, with the core meaning 'whole, uninjured'. This is one of the more revealing reconstructions to follow, because the same root surfaces in the Germanic branch of Indo-European along quite different lines and produces a small family of native English words: *whole*, *hale*, and *holy*. From a single PIE notion of intactness, Latin built *salvus* (safe) and a Germanic ancestor built *hailaz* (intact, sound), which in Old English became *hāl* — the source of *whole* and *hale* — and, in its sense of 'intact, set apart, inviolable', *hālig*, 'holy'. So *save* and *holy* are, at root, the same word.
Proto-Indo-European Roots
Within English, *save* expanded by sense rather than by form. Its earliest senses were physical and moral: to rescue someone from danger, to preserve a thing from loss, and — under the influence of Christian Latin — to deliver a soul from damnation, the sense that gives *salvation*, *Saviour* and *saved* in their religious uses. From there the word generalised. By the late medieval period it could mean 'keep in reserve, set aside', as in *save your strength*. The financial sense (*save money*) settled in early modern English, and *save* in computing — store data so it persists — is a twentieth-century specialisation of exactly the same idea: keep this from being lost.
The word has true Romance cognates wherever Latin *salvāre* survived: Spanish *salvar*, Italian *salvare*, Portuguese *salvar*, all meaning 'to save' or 'to rescue'. English *salvage*, *salvation*, *salvo* and *salute* are all from the same Latin root, borrowed at different moments and through different routes. *Safe*, the adjective, is a slightly earlier loan from Old French *sauf*, itself from *salvus*. Across all of these, the same image holds: to save is to keep something whole through time.