Upgrading an existing Azure installation is similar to upgrading a Windows Server installation.
Read the Upgrading an Existing Deployment section to familiarise yourself with the steps.
The main difference is that the files are copied up to an Azure file location via FTP.
Connect an FTP client such as Winscp to the Azure Website location.
The folder location on the Azure site is displayed in the Azure WebApp portal under Settings – Configuration – Path Mapping
Stop the Website in Azure using the Start / Stop button in the “Overview” section:
Ensure the web.config file has been edited locally to contain all the connection properties and settings of the version you are upgrading.
Using the FTP Client, rename the folder to something like “backupFoldername”.
Upload upgrade folder using same name in Path Mapping.
Click “Start” in Azure to start the website.