I love being able to see notifications of my tasks on the preview screen of my Android phone, but as of yet it’s not possible to mark them as complete there. Help us save a click! Let us mark a task as complete in the notification itself. 
I have the same question around marking tasks complete.
the guide section on email integrations makes it seem as though this feature actually already exists: https://asana.com/guide/team/onboard/asana-and-email#gl-email-integrations.
Manage email from Asana
Asana can also send you emails about new notifications. We recommend keeping this turned on until you’re in the habit of using Asana and checking your Asana Inbox daily. It’s also helpful if you simply work in email a lot, because you can take action on tasks from your inbox. Notification emails from Asana allow you to:Mark tasks complete—reply with Complete
Currently all the notifications I receive are from “no-reply@asana.com” – any pointers to what I’m missing here?