See: Turn Thunderbird into the Ultimate Gmail IMAP Client.
In your IMAP account settings, the Copies & Folders area, be sure to change the default location for Sent and Drafts to [Gmail]/Sent Mail and [Gmail]/Drafts respectively, as shown.
Even after you do this, you'll notice a few strange labels in your Gmail account: [Imap]/Sent, [Imap]/Drafts and [Imap]/Trash. These are Thunderbird's default Sent, Drafts, and Trash folders. Once you make the change to your account settings, you can delete those labels in Gmail and they won't get regenerated. (Note: except for [Imap]/Trash, which I can't rid myself of entirely, since T-bird seems married to it. Bueller? Update: see the next section for the solution to the [Imap]/Trash label.)