Send and Archive After Sending Messages in Mail

If you'd like an option to automatically archive messages after you've sent replies:

1. Click the gear icon and choose Settings.
2. In the General tab, locate the Send and Archive setting and choose the option you prefer.
    
3. Click Save Changes at the bottom of the page.

Once you enable the setting, you'll see a new button at the bottom of the page when you reply to messages. When you're done writing, click Send & Archive to send the reply and archive the previous message in one step, or click Send to send your reply but leave the original message in your inbox.

Reference - Google Online Help - Gmail - Archive