Adam Clayton had given Farsley a deserved lead inside 20 minutes, before James Lawrie equalised for the Dabbers in the second-half.
The visitors missed an array of chances and were made to pay as Ben Atkinson struck a late penalty to earn all three points for the Celts.
Nantwich struggled to cope with the blustery conditions early on as Farsley made a bright start, almost taking the lead through Adam Clayton as Will Jaaskelainen made a stunning reaction save to keep out the defender’s deflected effort.
Jimmy Spencer was the next to come close for the Celts, receiving the ball from Luke Parkin before shooting straight at Jaaskelainen.
The deadlock was then broken as Farsley’s Adam Clayton got the better of Jaskalainen, rising highest at the back post to head Luke Parkin’s corner back across goal and into the top corner.
Nantwich were then almost gifted a route back into the game as Jack Higgins’ weak back pass was intercepted by James Lawrie, but the Dabbers forward blasted his effort over the crossbar.
The visitors continued to press for the equaliser, and the lively Joe Mwasile was the next to come close, as he was allowed to turn on the edge of the box before curling wide.
It was the visitors who began to stamp their authority on the game after the break, and Sean Cooke twice came close to the leveller. Cooke’s curling effort from 25 yards out took a slight deflection past the upright. Minutes later, Louis Barnes’ cross was then deflected into the path of Cooke and he saw his effort saved from close range by Kyle Trenerry.
Joe Mwasile almost fired home a sensational strike just after the hour mark, running onto the ball 20-yards out and firing a sweetly hit strike inches wide of the upright.
Nantwich deservedly levelled 13 minutes from time, with Sean Cooke delivering a pinpoint cross for James Lawrie to dart in front of his man and head home.
Substitute Daniel Glover missed an excellent chance to complete the turnaround for Nantwich, as he managed to breach the Farsley defence, but his shot across goal drifted wide.
With five minutes remaining, Farsley were awarded a penalty as Dave Syers helped the ball onto Lewis Turner, who was brought down as he looked to cross from the right.
Substitute Ben Atkinson stepped up and rifled home a terrific spot-kick into the top left corner, which proved to be the decisive moment as the Celts earned a valuable three points.
Farsley Celtic: Trenerry, L. Turner, Hayhurst, Higgins, Ellis ©, Clayton, Syers, C. Atkinson, Spencer (Walker 74), Cartman (Baldwin 90+3), Parkin (B. Atkinson 67). Unused subs: Bower (GK), Bett.
Nantwich Town: Jaaskelainen, Morgan, Barnes, Stair, Mullarkey, Hughes, Mwasile (Glover 65), Bell, Malkin (Saunders 82), Cooke, Lawrie. Unused subs: Bourne, Cotterall, Fuller.
MOTM: Adam Clayton
Updated 01:19 - 13 Feb 2019 by Thomas Ritchie