Green’s Dictionary of Slang

to supply with a cigarette, to smoke a cigarette.

1918		Arbroath Herald 4 Oct. n.p.: After the band had bestowed its benediction in the strains of ‘Return to Serbia,’ the writer was fortunate enough to ‘fag’ an interpreter, through whom he sought out the native Scoutmaster.
1926		Maines & Grant Wise Crack Dict. 8: Fag me—Give me a cigarette.
1940		‘Smokers’ Sl.’ in AS XV:3 Oct. 335/2: To smoke is [...] to fag.
1954		(con. WWII) W. Faulkner Fable 361: ‘Fag me again.’ The corporal gave him another cigarette.