Thanks in advance.
Shakespeare's plays preceded Ben Jonson's song. And Shakespeare uses drink to with this meaning:
I drink to the general joy o' the whole table. (Macbeth)
With drinking healths to my niece: I'll drink to her as long as there is a passage in my throat and drink in Illyria: he's a coward and a coystrill that will not drink to my niece till his brains turn o' the toe like a parish-top. (Twelfth Night)
There is another definition listed in the OED that was current then,
To hand or present beverage for his use; to give drink to. Obsolete.
1470–1485, T. Malory, Morte d'Arthur: Thenne they lough and made good chere and eyther dranke to other frely.
However, this earlier meaning doesn't seem possible in Ben Jonson's song.