AFAIK there is no way to do this with threaded Gmail conversations. This is because the entire email "conversation" is connected to every other message, thus your sent messages will be removed.
Why does Archiving the email not work? Archiving will preserve all of the email threads in your account, remove it from the inbox view, and allow you to access things using the "All Mail" link in the left-nav. This will also preserve all of your Sent mail.
An alternative if you don't want to Archive email is to un-thread your email conversations. This will remove Gmail's threading functionality, making email behave more like Outlook where every email is treated as its own message. The upside is that you should be able to save sent mail separately from the entire email thread. The downside here is that it will no longer be able to group replies and forwards to the original email, making things more complicated.
To disable threading, just go to
Conversation View >
Conversation View Off >