Have you used custom logic on Submit button to save the form with auto generated name (EXp-FirstName-Now())?. If yes you need to change this logic. You can have one field (suppose FormName) which will not be shown on the form and is set when the form is opened for the first time. Use this field to name the form automatically when you submit the form. You don't need save option then. I have used same approach and it works fine. Make sure that Allow Overwrite option is selected.
Thanks,
Rahul Babar
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ASP.NET, C#, Sharepoint 2007/2010/2013/SP Online, Infopath 2007/2010/2013 Admin/Developer