Here is another way (that I use, but does not involve the browser):
1. Download (and install) a FTP client (I use FileZilla)
2. For 'host' enter the URL of your website.
3. Under username and password enter your cPanel username and password
4. Click 'Quickconnect'
These are the steps for using FileZilla. All FTP clients work approximately the same way.
Good luck on your website!