Within Teamwork Chat, you as a user have the option to receive important email notifications when marked "away" or "offline.
When you reply to an email notification from Teamwork Chat, your reply is posted to that conversation/room as if you were logged into Teamwork Chat.
To enable this select the settings cog to the bottom left of chat and then choose 'Notifications', you'll then have an option to toggle "Email notifications".
Here's an example of a sample email notification from Teamwork Chat where you can see who mentioned you, the conversation/room along with an option to open the conversation in chat.
For more information on notifications see: Notifications Settings