c. 1200, saven, "to deliver from some danger; rescue from peril, bring to safety," also "prevent the death of;" also "to deliver from sin or its consequences; admit to eternal life; gain salvation," from Old French sauver "keep (safe), protect, redeem," from Late Latin salvare "make safe, secure," from Latin salvus "safe" (from PIE root *sol- "whole, well-kept").
From c. 1300 as "reserve for future use, hold back, store up instead of spending;" hence "keep possession of" (late 14c.). As a quasi-preposition from c. 1300, "without prejudice or harm to," on model of French and Latin cognates.
To save face (1898) first was used among the British community in China and is said to be from Chinese; it has not been found in Chinese, but tiu lien "to lose face" does occur. To save appearances "do something to prevent exposure, embarrassment, etc." is by 1711; earlier save (the) appearances, a term in philosophy that goes back to ancient Greek in reference to a theory which explains the observed facts.
To not (do something) to save one's life is recorded from 1848. To save (one's) breath "cease talking or arguing in a lost cause" is from 1926.