There is a known problem described here ECP redirects to OWA in Exchange 2013 and here: Exchange admin center in Exchange 2013.
In short
If you’re in a coexistence scenario, where you’re running Exchange 2010 and Exchange 2013 in the same organization, and your mailbox is still housed on the Exchange 2010 Mailbox server, the browser will default to the Exchange 2010 ECP. You can access the EAC by adding the Exchange version to the URL. For example, to access the EAC whose virtual directory is hosted on the Client Access server CAS15-NA, use the following URL: https://CAS15-NA/ecp?ExchClientVer=15. Conversely, if you want to access the Exchange 2010 ECP and your mailbox resides on an Exchange 2013 Mailbox server, use the following URL: https://CAS14-NA/ecp?ExchClientVer=14.
So when I try the URL modification: https://<Exchange_2013_CAS_FQDN>/ecp?ExchClientVer=15
It shows up as OWA authentication but after you provide credentials it will redirect back to EAC.
The problem I have is that now I have to point and remind of this problem to an army of those who need access to ECP (used to create e-mail redirects). Is there a solution which could give me a break i.e. another workaround, more persistent than people memory? May be some IIS/Exchange setup tweak?