Behind to John Baird's first-half strike, the Lions looked to be in big trouble as they struggled to break down their lower-league visitors.
However, they were handed a lifeline when Rovers defender Iain Davidson was guilty of handball in the box with just 13 minutes remaining.
Lawless coolly slotted in the first of his two penalties before Lyndon Dykes hammered Livi into the lead just 60 seconds later.
They came even closer to the breakthrough 10 minutes later as they settled early following their winter-break lay-off.
Lawson was again the provider, this time with a cross from the right, but Dykes' bundled shot came back off the post.
It was Raith who grabbed the lead in the 22nd minute. The Stark's Park side won a corner on the left when Ross Matthews burst into the box but saw his cutback cleared before it could reach Baird.
Then, when Dylan Tait's flag-kick was only partially cleared, Baird was on hand to hook in Tony Dingwall's header back into the danger area.
Livingston bossed possession without creating too much before they cut loose late on to avoid an upset.
First, Lawless converted from the spot after Davidson's handball before Rovers coughed up possession straight from kick-off and Dykes drilled in from 20 yards out.
Time was now the enemy of Rovers and, when Michael Miller tripped substitute Erskine, Lawless eased Livi into the fifth round from the spot.