No, the Asana Outlook add-in doesn’t attach the email. I think the best thing one can do currently is drag and drop the email from the Outlook email list onto the task in Asana. But:
- That’s awkward because you have to go to the task in Asana which kind of defeats the purpose of having the add-in in Outlook
- Once in Asana, you can’t view the email directly; you have to download it and then open the downloaded .msg file
Here’s a request for this feature you can upvote: