UPDATE: This issue has been rectified in iOS 10 and later, as noted in the Apple Support link at the bottom of this page.
The Draft Folder syncs between OWA and Outlook via MAPI, but not with ActiveSync, IMAP, or POP protocol - which means that synchronization of drafts is not supported on mobile devices. When a message is composed from an account configured as ActiveSync on a mobile device, the message must be sent in order to sync with Exchange. Drafts created from the mail app are stored locally (on the device or mail client).
Apple has changed the way some of the folders are displayed since iOS 7, when drafts could sync down from the server, but not up from the device. On an iPhone 5 with iOS 8.1.2, the drafts folder cannot be seen by default. The folder can be configured to show up, but only local items will be displayed.
In order to view the Drafts folder, open Mail > tap Exchange > Mailboxes > Edit > See if Drafts (Exchange) is in the list and tap it to select it to be displayed.
If it is not visible tap Add Mailbox > Exchange > Drafts.
This will allow you to view, edit, and send drafts for your Exchange account that you created on the device. Unfortunately, drafts from the server will still not display.
This is a known limitation described on the following manufacturer support page: