I would like to suggest building a new Events for Office.JS library used in Web application hosted in Task Pane UI (Web Add-in) in Excel.
When End-user saves the Workbook the event gets fired. The event gets fired for the 'Save' and 'Save As' type of actions in the Excel workbook triggered by pressing a keyboard key combination or clicking on the 'Save' or 'Save As' button from the top panel ribbon or Workbook menu.
That way Task Pane UI using the Office.JS library can subscribe to these events and handle them.
Thank you, Martin