The visitors could have picked up all three points but for squandering a chance from the spot in the second half, as Max Dearnley stood firm to deny Tom Denton.
The first opening of the game went the way of Alfreton inside the opening two minutes, Fraser Preston glancing a header wide from a Dayle Southwell cross.
Both sides were largely restricted to half-chances in the opening quarter, with Bailey Hobson flashing an effort wide for the visitors, and Frank Mulhern heading wide for Farsley.
Alfreton were then perhaps fortunate to keep all eleven men on the pitch when Tom Denton clattered into Andy Butler, with Farsley claiming the forward led with his elbow.
That incident seemed to kick the Celts into gear, and they could have gone in front before the break.
A Jimmy Spencer flick-on found Adam Clayton who turned the ball inches over, before Spencer himself came close, striking the crossbar with an outside-of-the-foot effort from 18-yards.
Farsley picked up where they left off, starting brightly after the break; Spencer came close as he linked up with Luke Parkin before flashing a strike over.
However, after that, it was Alfreton who looked to kick on, and Max Dearnley had to be sharp to deny Bailey Hobson at his near post.
Billy Heath’s side then had a golden opportunity to break the deadlock, as Andy Butler was adjudged to have felled Dayle Southwell in the box. However, Max Dearnley guessed correctly, diving to his left to smother a weak effort from Tom Denton.
Alfreton would come close again at the death following an almighty goalmouth scramble, Dom Smith hitting the bar before Conor Branson and Denton both failed to apply the finishing touch from close range.
Farsley Celtic: Dearnley, Turner, Butroid, Johnson (Hornshaw 80), Butler, Clayton, Gratton (Malumo 86), Atkinson ©, Spencer, Mulhern, Parkin. Unused subs: Hutchinson, Morgan (GK), Ingham.
Alfreton Town: Willis, Lund, Wilder, Cantrill, Smith, Wiley, Preston (Dockerty 72), Branson ©, Denton, Southwell, Hobson. Unused subs: Whitham, Preston, Brown, Askew.