I have deployed my nodeJs RESTFull API with MySQL, on azure using Dropbox. On console i have seen logs that application started listing and database is connected. thus lines are
2 things:
You should use process.env.PORT
for handling the named pipe port in Azure App Service. See Listen additional port Microsoft Azure Nodejs.
If #1 didn't solve your problem, you'll need to enable logging of stdout and stderr for further troubleshooting. See How to enable BLOB-logging for a Node.js Api App on Azure?