This article explains how to get a Tableau filter action to open in the same tab. The default behaviour of Tableau Server is to open in a new tab in your browser. Using a Tableau hack, it’s possible to open the drill down worksheet in the same tab.
Firstly, how to create this issue of the action opening in a new tab. Note this article refers to a Tableau Filter Action, NOT a URL action.
How to make a Tableau Filter Action to Open in the Same Tab
Assume a workbook with 2 worksheets, a Summary and Detail. The user begins on the Summary and, using a filter action, they can drill down to Detail. Publish the workbook with “Show sheets as tabs” unchecked. The intention is the user navigates within the Tableau dashboard, instead of using the sheet tabs.
Set up a filter action to drill from the Summary to Detail and publish the workbook. Drill through from Summary and notice the Detail annoyingly opens on a new tab.

The hack to open in the same tab is simple. Float a Navigation Button on the Summary sheet. Set “Navigate to” to the Detail sheet.

Now hide the button. Float it on the dashboard, so it’s possible to set the XY coordinates and the set the size. Put it somewhere out of the way – such as X=1 & Y=1. Also size 1×1.

Publish the workbook, drill down from the Summary and see the Detail now opens in the same tab!
Credit for finding this hack goes to Athul Kalarikkal who posted it on the Tableau forum.
Download an example of this technique from Tableau Public.
Didn’t work for me. Same issue and the filter isn’t applied in the new tab.
Perhaps it depends on Tableau version? I tested on Tableau Online & Tableau Public (the same thing basically), and it worked there. Any chance you could mock up a demo on Tableau Public and see if you can get it working there?