Stoney struck in the first half at Poolfoot Farm with Lawson doubling the advantage before the break.
Lawson then delivered again in the second period to secure the win.
They are six points ahead of Fulwood Amateurs with one game to play, with Fulwood still having three games left to play.
GAMES LEFT:
Thornton - Blackpool Wren Rovers (a)
Fulwood - Tempest (a), CMB (a) and Hurst Green (a)
Jack Arrowsmith (pictured above) was on the scoresheet for Blackpool Wren Rovers but Lytham Town won the local bragging rights with a stunning 3-2 comeback in the Fylde derby.
Wrens looked in control at 2-0 through goals from Alex McLaughlin and Arrowsmith.
But Lytham hit back for a memorable 3-2 win thanks to Bob Harris, Danny Scarlett and Lloyd Patten.
Hurst Green and Tempest United played out a 0-0 draw in the other fixture last night.
In Division One Reserve, bottom of the table Eagley grabbed their first home win of the season with a 4-1 defeat of champions Euxton Villa Reserves.
Sam Smedley, Rhys Hardie, Cole Collins and Alfie Cowen grabbed the goals, with Jacob Greenhalgh on the scoresheet for the visitors.
A Ben Davies double helped Lostock St Gerards to a 2-0 win over Croston Sports in the other match played last night.
WEDNESDAY PREVIEW
In Division One, Kendal County will be looking for maximum points at Lostock St Gerards in their bid to stave off relegation.
They are three points behind Horwich St Mary's with two games left, with the Bolton side having their final match this Saturday.
Garstang Res and Burnley United complete their seasons in Division Two while Turton Res host Horwich St Mary's Res in Division One Reserves
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