The phrase seems to be of Scottish origin. As Hugo found, most of the earliest uses of the phrase have to do with alcohol consumption. I did find this earlier figurative use of the phrase however from The History Of The Church And State Of Scotland, 1753 (date check):
It appears the figurative sense of under one's belt to mean owned or "contained by" goes back even further as evidenced by this old Scottish saying from A Complete Collection of Scotish Proverbs, 1721:
