Indeed there is no Rest API request for this. One way to do it would be to create the task and run it in one go.
Such as Create Task -> Query it -> Run it.
You can also suggest the implementation of this in our ideas section
What version of Tableau server are you using. I found this method that you could use.
Update Datasource Now
Runs an extract refresh on the specified data source.
This method runs an extract refresh for the specified data source, with no need to associate that extract refresh with a scheduled task. This method is the equivalent of selecting a data source using the Tableau Server UI, and then selecting Refresh Extracts from the menu (also known as a "manual refresh"). Similar to a manual refresh, if the extract is a
.tdefile, it will be converted to a
.hyperfile as a result of the refresh. To learn more about extract upgrades to the
.hyperfile format, see Extract Upgrade to .hyper Format.
This is the REST API for 10.5 though. https://onlinehelp.tableau.com/v10.5/api/rest_api/en-us/help.htm#REST/rest_api_ref.htm#Update_Datasource_Now%3FTocPath%3DAPI%2520Reference%7C_____94
Hope this helps.