comparative of high (adj.), Old English hierra (West Saxon), hera (Anglian). Higher education is attested by 1839.
The French distinguish l'instruction secondaire, which includes what we term a liberal education, from l'instruction supérieure, which denotes professional education; but I do not think the corresponding English phrases are used with this distinction. [William Whewell, "Of a Liberal Education in General," 1850]
Higher-up (n.) "one in a superior post" is from 1905, American English.