Yes, you can. But only if the replays are not older than 15 minutes.
By using the command "/alts" in a chat room or PM, you can view matches that are earlier than 15 minutes.
This is the only way to view your battle history for unsaved matches. Under "recent games", you can view all of your matches played under 15 minutes.
This way, you can view unsaved matches then upload the replay/download it.
Hope this helps.