The question is already answered but I think I've found the solution that is simpler and more relevant to the question title, here it is:
var cli = new WebClient();
cli.Headers[HttpRequestHeader.ContentType] = "application/json";
string response = cli.UploadString("http://some/address", "{some:\"json data\"}");
PS: In the most of .net implementations, but not in all WebClient is IDisposable, so of cource it is better to do 'using' or 'Dispose' on it. However in this particular case it is not really necessary.