Livingston failed to record a home win in the League for the second time this season as leaders Partick Thistle maintained their 100% record in a narrow win.
The visitors had the best of early opportunities as Lawless tested McNeil with a dangerous cross ball, and then a Callum Fordyce deflection turned Stuart Bannigan's cross goal wards, but the stopper was sharp to flick the ball to safety.
Iain Russell was presented with the home side's best opportunity of the half when Barr's long ball sent him clear, but the striker's lob sailed wide of the target.
Aaron Muirhead was next to threaten for the visitors, but his effort from an lan Archibald lay off went over.
Lawless ensured the visitors got the goal they deserved however, as he smashed home from 20 yards.
Thistle could have made it two seconds later, but Chris Erskine could only hit the post after being teed up six yards out.
Just before the whistle Morton nearly dragged Livi level with a fine effort from outside the box, but the ball smacked the post and spun out to safety.
It didn't take long for Thistle to double their lead after the break. lawless again was on the end of things as a fine move opened Livi up down their left.
The home side refused to let Thistle cruise home however, and after Craig Barr bundled the ball to Morton from a Bobby Barr corner, the winger blasted home by way of a deflection.
Marc McNulty almost grabbed a point for Livi as his low shot was well saved by Scott Fox on 77 minutes, but it was too little too late as the Jags held out to win all three points.
The result left Livi with two defeats from two in the league here in West Lothian, while Thistle secured their second away win of the season after defeating Dunfermline.
McNeil, Fordyce (Easton 81), Booth, Garcia Tena, C.Barr, Docherty, B.Barr, I.Russell, Andreu, Scougall (McNulty 61), Morton
Subs not used: McDonald, McCann, Jamieson
Fox, O'Donnell, Muirhead, Archibald, Bannigan, Paton, Murray, Welsh, Erskine, Lawless (Sinclair 85), Craig (Elliot 9)
Subs not used: , Balatoni, Craigen,
Referee: Craig Thomson
Official Attendance: 1,745
Match report: Calum Leslie
Pictures: Steven Carrie