I will try to summarize the support article here
Recipe:
Ingredients:
1 Azure app service
1 Domain name with a non Azure DNS provider
Step 1) Take note of the Azure Url. ( App services-> selected app ->Overview->URL ( in the top right)
Step 2) Take note of the App Service IP Address. (App services->selected app->Custom domains-> I P Address)
Step 3)
Add the following records in the DNS
Add a CName record for www with alias mywebsite.azurewebsites.net (i.e from step1)
add an A record with the IP address ( from step 2 )
add a TXT record with alias mywebsite.azurewebsites.net (from step1 again)
Step 4)
App Services -> Selected app service-> Custom domains-> Add hostname
enter mywebsite.com ( or what ever it is)
click Validate
click Add Host Name ( if the validation is successful)
Step 5)
Repeat step 4 for www.mywebsite.com ( not the real name)