You could use Azure AppService to host your app. With this service you can get a free app if you use their subdomains and very little resources. Pay U$$ 10 per app if you want custom domain but still using little resources.
If your app uses a reasonable amount of resources (cpu, ram, etc) then you should use basic at least, this cost $55.80/mo and you can host as many apps as you want (your limits are on cpu / ram)
This last option may sound expensive but we have advantages here. The advantage of using AppService is that you don't have to manage things like windows server updates, security, etc. Also you gain the hability to do deploys using git which saves a lot of time (and time is money). Furthermore you have development slots to test different versions of your site before going to production.
As for the DBaaS they have Azure SQL Database which is API compatible with SQL Server. As far as most application that uses SQL Server works, is a direct replacement. You just change the connection string and it works. We have here also the advantage of not having to install updates or secure sql server. You can even left Azure the backup thing (though you can also manage it yourself). You get point in time restore which means that you can restore to any point in time (up to some limit depending on service tier). This can be use to solve human and software errors, or hacks with only a couple of clicks instead of dealing with backup files and remoting servers.
Pricing for DBaaS Azure SQL Database is per performance basis and uses a thing called DTU (Database Transaction Unit). You can use Azure SQL Database Query Performance Insight to measure how many DTUs you need but a basic service tier runs from only U$S 5.
We have to remember that with cloud we are paying per hour and we can scale up or down resources if we need to save more money or gain more performance.
Prices are for Central US Datacenter. With others it may vary. Also if you have a Visual Studio Subscription (previously MSDN) or Bizpark or Action Pack you can get some discounts.
In order to know how to better use your money in Azure Cloud I can recommend you this article by Scott Hanselman. In general the Penny Pinching Series by him is a lot resource to learn and understand more about this.