Need to change submitted form entry data when a customer input wrong data.

You may edit the form submissions from the submissions view page by clicking on Edit button.

https://www.jotform.com/help/269-How-to-View-Form-Submissions 

Hope this information helps! 

I am aware of that, but there is no save option. After I make the change of the customer input mistake, all that is available is a close option and that does not save the change.

Do I have to use the submit button in the form for it to save?

Yes. While editing a submission, it is required to submit the form again and that is how the modified information will be stored to the submission.

Hope this information helps!