To automate the process of saving new JotForm responses as rows in a Google Sheet, users can set up a Zap using Zapier, a workflow automation software. The process involves creating a header row in the Google Sheet and setting up a trigger to connect with JotForm, where the user collects customer data such as feedback or survey responses. Once the connection is established, the user sets up an action to add the collected data to the Google Sheets rows, mapping the relevant fields from JotForm to their corresponding columns in the spreadsheet. The process can be tested and validated by sending a sample response through the Zap, ensuring that everything works as intended before using it for actual business purposes.