The winger struck after a long throw into the Motherwell box in the 22nd minute and Livingston rarely looked like relinquishing their lead at the Tony Macaroni Arena.
Home goalkeeper Liam Kelly made his first meaningful save in the 79th minute and Craig Halkett settled matters four minutes later when he headed home from a corner.
Motherwell had an opportunity to move four points off the top six after winning three of their previous four matches but they were indecisive in defence and lacking intensity and purpose in their attacking play, and they saw local rivals Hamilton move within a point of them.
Livingston had not scored for four matches and saw stand-in striker Scott Robinson drop to the bench with a hamstring problem on top of the absence of the suspended Dolly Menga and injured Lee Miller, Rafa De Vita and Ryan Hardie.
Teenage striker Jack Hamilton made his first start since scoring in manager Gary Holt’s debut at St Mirren in August.
But Motherwell had selection challenges at the back with Carl McHugh and Peter Hartley suspended and Charles Dunne not ready to return from a groin problem.
Inexperienced pair Christian Mbulu and Barry Maguire lined up either side of Tom Aldred but the back three were all guilty of making sloppy passes which helped put their team under pressure in the opening stages.
Motherwell had a let-off when Mark Gillespie made an outstanding point-blank save from Lawless following a long throw from Alan Lithgow.
But they did not heed the warning and Lawless squeezed a shot past Gillespie from 14 yards after a similar move.
Hamilton could not quite get hold of a couple of shots from decent positions but Livingston remained comfortable as Motherwell struggled to get any passing rhythm going.