Emulator's got this pretty much correct.
There are various ways of achieving this effect, and back in the day, it depended on which gear you had, and in the case of Ultravox, some custom made equipment.
I believe Warren controlled the rhythm of the trigger pulses from a custom unit that was synchronised to the clock signal from the drum machines (initially Roland TR77 and CR78 on Vienna and RIE, then Linn LM-1 and LinnDrum on Quartet, and SCI Drumtracks on Lament). The custom trigger unit had 16 switches which corresponded to 16th notes. Whichever switches Warren switched on would send a trigger pulse, and the ones not switched on would equate to a "rest".
I've seen photos of at least two versions of this trigger unit. The early version was in the article in the UK magazine "Electronics & Music Maker" which featured Warren talking about this setup. I also saw a later pic somewhere which had an altogether more "pro-looking" unit badged "Cactus" (the company who made the short-lived "Desert Drums" electronic drum kit which was a Simmons kit-a-like).
The Pete Woods trigger router allowed the trigger pulses to be sent to any of the synths, either Chris' MiniMoog or Midge's or Billy's.
Chris then had to play extremely tightly to ensure that he had the right note pressed down when the trigger pulses gated the envelopes on the Mini. This takes quite a lot of skill and discipline to do.
The best way to do this with "standard" kit is to use a Roland drum machine to generate the triggers - TR808, TR909, TR707, TR727, TR606 are the best, as they work in exactly the same way, but are pretty expensive these days. Some of the newer Rolands still had analogue trigger outputs, but lost the intuitive "16 switches" approach, and replaced with an LCD programming grid (see TR505, TR626, DR110 etc), and these would be OK. You then need an analog monosynth with an analogue trigger / gate input. Emulator mentioned the SH101, but SH09 / SH2 would similarly do it. Most analogue monos from that period would work. The trigger out from the drum machine then "gates" the synth, and the note pitch is determined by the key you hold down.
The alternative approach, which will work with any synth - including poly synths - is to use a noise gate, where the trigger pulse goes into the sidechain input, and the synth goes in the audio input. The net effect is that the sustained notes or chords you play on the synth are "gated" by the trigger pulse rhythm from the drum machine. The controls on the noise gate effectively then become like the ADSR on the synth.
I believe the latter effect was also used by Midge & the boys on tracks on Quartet, although I can't be sure - it could have been done on the PPG 2.2 poly sequencer. Same effect also used on the excellent Midge-produced track "Together" on The Phil Lynott album (one of my personal faves).
Of course, as previously stated - today, most people would just sequence it in Logic, Cubase, Reason or Sonar, but that's not so useful if you want to play it live (although you could set up Reason to do this live).
There are a number of newer synths out there that have capabilities to play short sequences from single notes - I think the JP8000 calls them "Realtime Phrase Sequences" RPS. I think my old Korg Prophecy can do it too, and I think the Yamaha CS1X might have something similar, though don't buy without checking, as my memory's not my best feature!
Hope that helps.