I have created a node.js application in Visual Studio 2015 using the Azure SDK 2.7 and the Node.js Tools.
I have successfully set up CI with BitBucket in my web app
Edit: Try creating "Node JS Empty Web App" from the gallery at https://portal.azure.com and compare the web.config and the site with what you have. It's possible that you're missing some config settings.
Previous answer: first off, only ports 80 and 443 are available in Azure Web Apps (the new name of Azure Websites). So port 1337 will not work. Reconfigure your app to run on port 80 or 443. Regarding the permission issue, do you have App Service Authentication enabled? Make sure that is disabled, by editing the Web App's application settings as below.