There is a way you can use your web browser for FTPing files from your local computer to your webspace.
If you go to Domains > Manage Domains, you will see "FTP" by the name of your domain.
Click on it and you will be prompted for your FTP password. Once you're logged in, it will be like you have a folder opened up on your desktop. You can then drag and drop into that folder from other open folders on your computer.
I have done this in the past using Internet Explorer (not my usual browser) and it seems a little slow, but it works.