Green’s Dictionary of Slang

drunk; thus put under, to render drunk.

1939		H. Miller Tropic of Capricorn (1964) 68: The only thing they had in the house was Kummel and it didn’t take much to put us under.
1940		(con. 1918) S.J. Simonsen Soldier Bill 125: English ales run well over ten percent and a person wants to be careful not to drink too much as it can easily ‘put you under.’.
1967–8		Baker et al. CUSS 216: Under Drunk and passed out.