A 79-point deduction for fielding an ineligible player sent Follifoot to the bottom earlier in the season but they have clawed their way back and are only one point behind Ilkley, the side just in the safety zone.
Victory looked unlikely when Follifoot slumped from 120-6 to 130 all out but Lawrence kept Rawdon on the back foot.
The final pair of Stuart Ward and Harry Nicholson edged towards the target, putting on 11 before Alex Fox trapped Ward lbw.
With every point between now and the end of the season vital to survival hopes, Follifoot’s Nick Robinson was impressed with the win.
He said: “The win was crucial.
“Every game now is vital and we need to keep picking up points and putting pressure on the teams above us.
“Ilkley and Guiseley will be fighting as hard as us to stay in the division and we need to pick up more points that them from the remaining games; it’s as simple as that.
“With the points deduction we were given we set a target of staying up and we certainly believe we can do it.”
Ilkley went down to defeat at Kirkstall, who made 217-9 with Stephen Wilkinson scoring 98 and Henry Wilson taking 4-48.
Stephen Brown (5-42) and Babar Butt (4-61) made sure Ilkley could not challenge but they got to 175 and a crucial third batting point.
James Davies (67) and Jamie Pickering (48) led the way as leaders Otley made 184 before Ricky Halloran (5-11) and Sam Blake (3-25) dismissed Guiseley for 104.
That keeps Otley on top and sent Guiseley to the bottom.
Second-placed Beckwithshaw beat Collingham to stay hot on Otley’s heels as the pair head towards a final week showdown.
Ben Quick (84) and Miles Buller (53) shared a century stand in Beckwithshaw’s 213-8 and then Yasar Ali took 6-35 as Collingham slumped to 134.
Having beaten Burley in the Waddilove Cup final, Bilton completed a league double over them thanks to Ryan Bradshaw (77) and Sajad Ali, who followed up a half-century with 5-33.
James Emmott looked to have put Steeton on the way to victory when his 5-39 helped restrict North Leeds to 150.
But Aftaab Karim (5-38) and Larry Audain (5-27) countered and the visitors were dismissed for 92.