There's also a service called Azure ACI which is pretty much the same thing as fargate
(disclaimer: I work for Azure on ACI and other container offerings)
Azure Container Instances (ACI) is what you're looking for. The intent behind ACI is to provide the easiest/simplest way for you to deploy serverless containers in Azure (no management of underlying infrastructure). See this to get started: https://docs.microsoft.com/en-us/azure/container-instances/c....
Not sure there is a direct equivalent but Azure App Service will let you deploy a docker image and autoscale. The configuration is more analogous to AWS Elastic Beanstalk but they serve similar purposes.
If anyone has any feedback or advice I'd happily hear it.