If you don't mind a little VBA, this macro will save before closing:
Private Sub Workbook_BeforeClose(Cancel As Boolean)
ActiveWorkbook.Save
End Sub
to install it go to the menu bar (File, Edit, View...)right-click the tiny
Excel icon just to the left of File. Select View code and paste the macro
into the open VBA window. Then just close the VBA window.
Once installed in a workbook, you can edit away. If you do a
File > Exit
or
File > Quit
The save will be automatic (no prompt at all)