I had the same problem. The windshield washer pump was busted on my 01 TT. Just replace the pump and it should be fine. It's a $35-60 piece depending on who you buy it from (dealer is more like $60), but replacing it yourself can be kinda tricky.
Edit:
http://www.audi-forums.com/1998-2006...sher-pump.html
This was my thread on the issue. Thanks goes to Stamina for the info!