I just talked to the support and they confirmed that currently its only possible to save email attachments within Salesforce by using the Outlook plugin OR doing manually from within salesforce.
I strongly suggest to add this functionality when using the "email to Salesforce" option, for example when sending mails from Googe Apps (Google Mail) and also when sending a mail from within Salesforce.
There is truly no reason why this should not work with these two channels, especially if you have chosen "Always save email attachments" in your "My email to Salesforce" options.
Please let me know if you are working on solving this issue already.
Thanks and regards,