OK, we shut down our session last night on a roll the Outlaw Tech made because A) We'd been playing for 4 hours and I had stuff to do early this morning B) Interpreting this roll was giving me a headache because the three players were getting argumentative about it.

More or less he rolled both a Triumph and a Despair while trying to shutdown a security system using a computers check. The roll overall succeeded by one with no Advantage or Threat, but from what I can tell from reading the rules the Triumph and Despair don't cancel out their secondary effects. My initial take, and I still think this would work, is due to the Despair he sets of ALL of the alarms. Fire, reactor leak, intruder, all the klaxons, all the lights you name it and it's going off. But due to the Triumph it will be responded to as a computer glitch since his intrusion into the network wasn't detected. At most, only one or two security personnel will come to make a cursory patrol in the affected area just in case.

Can I get another take on this please?

That's not a bad interpretation and jeez, I don't envy being in your shoes with Triumph vs Despair! Hm, since it's still an overall success, I would do what you suggested, but the security techs decide that they need to shut down and reboot the whole security system, giving some time interval when no security measures are in place - i.e. locked doors will be unlocked, cameras won't be recording, motion-detectors won't be working, no defensive slicers will be monitoring the intranet, etc.

The Despair aspect is the initial alarm and the limited time of the system being down, but the Triumph + Success is that the system will be down for some time and the alarm won't be viewed as "legitimate" but as a system glitch. So an overall win for the players, as long as they get in and out in time.

Whenever you get Triumph and Despair on the same roll, check one thing first:

Did the check succeed or fail?

That's the first thing you need to decide. Whether or not the PC succeeded at what he was attempting is key to determining what went wrong.

As the GM, I would determine the Despair result first. So, in this case, setting the alarms off on a (still successful) computer intrusion is totally valid.

Then, let the player suggest what he wants to do with his Triumph. If he doesn't have any ideas, suggest some, but let the player think of something first, since it's his moment of (literal) triumph, after all.

Sounds pretty reasonable to me.