Because it hinges on the wine (4)
'the wine' is the definition.
'because it hinges' is the wordplay.
'because' becomes 'as' (I've seen this in other clues).
'it' becomes ''t' (abbreviation. e.g. in 'tis).
'hinges' becomes 'i' (I can't justify this - if you can you should believe this answer much more).
'on' is the link.
Can you help me to learn more?
(Other definitions for asti that I've seen before include "Italian product", "18", "Piedmont wine city, SE of Turin", "Bubbly wine", "Sparkling white wine".)