I have tried adding DefaultConnection from my appsettings.json file to Azure\'s Application Settings but Azure will not override the connection string.
I just had a similar problem (the problem was with PostgreSQL connection string type, I had to change it to custom) and now it works for me, so these are the pieces:
Additional info from my comments below:
Azure GUI (Connection strings, Application settings) uses environment variables internally, so the appsettings.json will stay the same.
If there is a need for an appsettings.json's value to be overwritten during VSTS release activity (before it will be published to Azure), Colin's ALM Corner Build & Release Tools can be used. Here are the links to Colin's ALM Corner Build & Release Tools and tutorial.