You can, but it's a bit tricky. Use the InternetSetCookie call together with a CookieContainer from HttpWebRequest.
Here's the method: http://msdn.microsoft.com/en-us/library/windows/desktop/aa385107(v=vs.85).aspx
and here's an example on how to set it: http://social.msdn.microsoft.com/Forums/en-SG/csharpgeneral/thread/76a38eee-3ef6-4993-a54d-3fecc4eb6cff