The API supports status, when I set it for example to Duplicate I get the same results as when I filter via the WebUI.

https://pagure.io/pagure/issues?status=Open&close_status=Duplicate
https://pagure.io/api/0/pagure/issues?status=Duplicate

Can you please give an example what's not working for you so I can better understand and reproduce it?

Based on the webUI URL parameters I thought that status and close_status were also to be used separately in APIs and that close_status wasn't available in APIs. It is not clear (to me) in the API docs that I could have used close_status values for status, I thought I need to use something like
https://pagure.io/api/0/pagure/issues?status=Closed&close_status=Duplicate

About the second part of my message, it seems odd in the webUI URL to use status=Open together with close_status... after your reply I've just realized that when close_status is present in the URL, status is just completely ignored. For example these are identical:
https://pagure.io/pagure/issues?status=Open&close_status=Insufficient+data
https://pagure.io/pagure/issues?close_status=Insufficient+data

And, oddily, starting from the URLs above if you change the close_status from the dropdown filter you'll end with something like
https://pagure.io/pagure/issues?status=Insufficient+data&close_status=Invalid

Thanks for your Feedback @mattia, let me put that on the Milestones for the next 5.13 Release, it's partially a documentation thing but also there seem to be some bugs in the UI depending on what's selected, requires some time to dig deeper.