The Dutch forward crashed in a 13th minute free-kick and then glanced in a header on 69 minutes to keep Graham Alexander’s side in fourth place heading into the winter break, despite Bruce Anderson's late consolation strike for Livi.
With ’Well’s top scorer Tony Watt on the bench amid reports he has agreed a pre-contract with Dundee United, the Steelmen took to the field in front of a minimal crowd due to newly introduced coronavirus regulations restricting crowds at outdoor events to a maximum of 500 people.
Those who were inside a freezing Fir Park witnessed a first half of few chances other than van Veen’s brilliant strike, which he struck into the top right corner following a foul on Nathan McGinley.
Other than that, Watt’s replacement for the day Connor Shields shot over from close range, albeit the whistle had already been blown, and Sean Goss curled an effort straight at away keeper Max Stryjek.
Motherwell wasted a magnificent opportunity to double their advantage on the hour mark when van Veen played in Goss on the left hand side, but Stryjek denied him with a low down save.
But the away custodian was powerless to stop Motherwell claiming what turned out to be their decisive second goal.
Shields’ fine cross to the back post was nodded in by van Veen to send the few ’Well supporters who were inside the stadium wild with delight.
Fir Park forward Kaiyne Woolery then had the chance to put the hosts out of sight but he couldn't control the ball properly after a fine van Veen cross.
The last few minutes were made interesting when Anderson netted a James Penrice cross, before van Veen almost sealed his hat-trick in the dying minutes.
He rampaged forward from the halfway line and sent an attempt goalward which Stryjek managed to keep out.
Motherwell, whose total of 31 points from 20 games puts them two clear of fifth placed Hibs, have a break until their next league game against Ross County in Dingwall on Tuesday, January 18.
Motherwell: Kelly, O’Donnell, Mugabi, Solholm (Ojala 79), McGinley, Donnelly, Maguire, Goss, Woolery (Roberts 90), van Veen, Shields (Grimshaw 84).
Livingston: Stryjek, Fitzwater, Obileye, Longridge, Kelly, Sibbald (Anderson 60), Holt, Omeonga (McMillan 74), Penrice, Forrest (Panayiotou 71), Shinnie.
Referee: Don Robertson