Is there an Express version of Azure SQL DB? With on-premise SQL Server, there are editions such as Express, Web, Standard, and Enterprise. Do those editions (or an analog) exist in Azure SQL DB, especially Express? 

My Google-Fu is failing here, possibly because it doesn't exist.
Azure SQL DB does not have a direct counterpart for SQL Server Express. 
Some functionalities that exist in SQL Express or other versions are simply not there, like filestream. (Also not supported on managed instances). 

As a result, you need to find the minimal functionalities you need and see if Azure SQL DB is fit to be a replacement of your SQL Express.

To do this, you would have to look into the Azure SQL DB Purchasing Models.

There are some differences in functionalities depending on the azure tier, as @AaronBertrand mentioned:


  There are some functionality differences between tiers and service
  levels. As two off-the-cuff examples, Columnstore is only in S3 and
  above and In-Memory is only available in Premium and Business Critical
  - you can spot other differences in this doc


As some of these functionalities that are present in higher tiers are available in SQL Express Edition, you will have to check for these and find the minimal tier you could use. 

After finding the minimum tier you could use, all that is left is the sizing of your database, the DTU's and other aspects like max concurrent connections.

This all depends on your pricing levels.

At the time of writing, and solely based on the DTU-based model, the pricing levels are: Basic tier, Standard tier, Premium tier and PremiumRS tier.

The vCore-based model also exists, but (again, at the time of writing) there is no low cost option like Basic or low DTU standard.

vCore-based model tiers

Comparison

Basic tier

Basic tier might be what you are looking for. 
The downside is that it only goes up to 2GB. 
It is not free like SQL Server Express edition, but it will be the lowest cost, granted that:


you are able to stay under 2GB
you don't need any of the other functionalities
a maximum of5 DTU's is enough.
...



Basic tier



Standard tier

If one or more of the above conditions are not met, then you will have to at least go for the standard tier. 

It will be more expensive, the price depending on how many DTU's you are going to need (S0 until S12). You can go up to 250GB in this tier.


Standard tier



As a sidenote, there might also be functionalities that you have or use in SQL Server Express which are simply not available in Azure SQL DB, such as filestream. (Also not supported on managed instances). As a result, you need to find the minimal functionalities you need and see if Azure SQL DB is fit to be a replacement of your SQL Express.