I have a simple mvc web site (using the VS internet template) on azurewebsites talking to a SQL Azure database in the same data center. The database at this time is just to
Add a <clear/>
element as the first element under the connectionstrings element to ensure no strings are being inherited. Also, I assume your Azure SQL DB is set up with proper firewall settings allowing your IP where the ASP.net code is deployed?
I also had this error. But in my case I did not notice that the connection string was being overwritten by the publish wizard. I fixed the problem by going to the Settings tab in the Publish wizard and unchecking "Use this connection string at runtime (update destination web.config)"