Which of the following sentences is correct?
1) I drank three too many beers last night.
2) I drank three beers too many last night.
Both. The main point is that you drank too many beers. Where you put the quantifier (three) or the identifier (beers) doesn't change that in this case.
That said, without the "three" you would not normally use the phrasing "beers too many."
For example, if someone asked you to drink all the beer(s) on the table (say there were 30), you might say: "That's too many beers ("to drink" being implied)," but you wouldn't say "That's beers too many." But you might say: "That's 25 beers too many."