Darren Stephenson’s close-range finish mid-way through the second half proved to be the only difference between the two sides in their first meeting in over five years.
Farsley’s on-loan goalkeeper Kieran Flavell produced a sensational second half performance after pulling of a string of saves as Bamber Bridge took the ascendancy, ensuring the Celts progressed.
The hosts were quickest out of the traps and were almost ahead after just eight minutes when Jack Baxter’s powerful volley struck the upright.
The Preston-based side continued to look dangerous in the opening quarter of an hour with Mark Cullen then drawing a good save from Flavell before Michael Potts headed just over the target.
However Farsley got a foothold in the contest and struck the opener on 22 minutes when a corner was played into the box and Conor Branson’s strike flicked off Darren Stephenson who got the final touch to fire home from just a couple of yards out.
The Celt Army had a couple more chances to extend their lead during the first half with Ben Atkinson and DJ Campton-Sturrdige’s efforts both thwarted by James Pradic in the hosts’ goal.
Potts came close to levelling just before the half-time whistle but Flavell got down well to his near post to deny the Bamber Bridge player.
Just two minutes into the second half, a great move saw Stephenson square to Campton-Sturridge but the forward was unable to beat Pradic who made an excellent save to keep the deficit at one.
As the game passed the half hour mark, the Preston-based side began to dominate and Flavell made key saves to deny Chris Churchman’s 25-yard free-kick on 67 minutes before the on-loan Barnsley keeper denied Potts from point-blank range.
The 21-year-old shot-stopper then produced an excellent save to thwart Jack Baxter before another unbelievable stop to deny Potts for a fourth time in the contest.
The Northern Premier League side continued to apply the pressure late on with a string of attacks and corners but stout defending from the Celts saw them scrape through the tie to progress to the Third Round.
Farsley Celtic: Flavell, Hyde (Watson 61), Branson, Misambo, Pickles, Sloane (Pickard 46), Fielding, Turner, B Atkinson, Campton-Sturridge (Fox 69), Stephenson.
Subs not used: Mampala, Youmbi.