JSON is available in all service tiers (basic, standard, and premium) but only in new SQL Database V12. You can see quick introduction here or more details in Getting Started page. you can also find code samples that JSON functions in Azure Sql Database on official Sql Server/Azure Sql Database GitHub repository.
Note that OPENJSON function requires database compatibility level 130. If all functions work except OPENJSON, you would need to set the latest compatibility level in database.
It will be interesting to see adoption of JSON within Azure SQL Database. I could see it being a bit more likely due to DocumentDB.