Eric Partridge, A Dictionary of Slang and Unconventional English, Fifth Edition (1961), says that the term originated (with the meaning "slightly drunk" circa 1873, and that by 1880 it had expanded to include the meaning "drunk in any degree." Partridge says that the term probably derives from "skew-whiff, perhaps on swipey." Here are the entries for those two terms in the same dictionary:
skew-whiff, adj. and adv. Crooked(ly); askew: dial[ectal)] and coll[oquial]: 1754 (S[horter] O[xford] D[ictionary]) —2. Hence, tipsy: C. 20.
swipey. (Not very) tipsy: coll[oquial]: 1844, Dickens, 'He's only a little swipey, you know.' Neveral gen[eral] and, by 1900, ob[solete].
Swipes was a late-18th-century and 19th-century slang term for beer.
I first heard the term in the 1960s, thanks to the invaluable (to a citizen of the United States) vinyl recordings of Beyond the Fringe. In a skit called "Aftermyth of War," an aristocratic lady of advanced years recalls the outbreak of World War II:
I turned to my husband, as he then was, and said "Squiffy, this is the end of an era."