Hello Kats,
It is possible to connect to Cloud SQL instances from external applications which includes Microsoft Azure as described in this document.
This is possible to connect to the Cloud SQL instances from external applications by using a proxy, by configuring access for one or more public IP addresses, by using the JDBC Socket Factory which is applicable to Java Programming language and MySQL Second Generation instances only and by using the Cloud SQL Proxy library which is applicable for the Go programming language and Second Generation instances only.
Please also note that if you have not created a Cloud SQL instance already you may encounter issues during creation as there is a Global issue currently ongoing which you can monitor here. Once the issue is resolved you can attempt to create the Cloud SQL instance and connect with the Microsoft Azure App Service application.