If you have telnet access to the unit you can do a scripted shutdown - and it may be possible to achieve the same thing without telnet - please note - I have not tested this with recent versions of the firmware.
From a command line prompt use touch to create a file called shutdown in the /tmp folder - touch /tmp/shutdown - will do it, and the unit will shut itself down. This used to be a part of the front panel shutdown process, but like I say, it's been awhile, so I don't know what changes have been made in the newer firmware versions.
I don't think it matters how you create the file, or even the contents of the file.
Now there are challenges to running telnet from a script, and out of the box there is no telnet server running on the DNS-323 - trust me, it can be done, Google is your friend - I used to have a script running on a Windows server that ran Powerware's Netsafe shutdown software - Netsafe would call the script before triggering the Windows shutdown routines and the Windows server would shutdown, the DNS-323 would shutdown and then the UPS would turn itself off.
APC's PowerChute also has similar capabilities, so you should be able to achieve similar results.
A second way to go about this would be to modify D-Link's NUT (Network UPS Tools) implementation - the DNS-323 in a couple of the newer firmware revisions supports UPS monitoring using what appears to be a default NUT implementation. NUT, as the name suggests, can communicate over a network and at least in theory, can be used to shut the unit down on command.