Our organisation recently switched to Office 365 and I see very nice integration with Asana available in Outlook web-mail. However I cannot find a way how to make it work - every time I want to comment or complete task I get message that I need first to authenticate with Asana. Although I did this authentication and Asana Add-In on right side is functioning.
Finally I solved it myself. I had to use Outlook Desktop and authenticate Asana through it (Outlook Web version did not worked). Then all integrated buttons started to work both on Desktop and Web versions of Outlook.