The hosts took a 17th-minute lead through Owen Story and while Paul Walker levelled one minute after the hour mark, Story then netted again within the next 60 seconds. But Farsley’s Alexander Davidson struck four minutes before the break to level the tie at 2-2 and Nightingale then netted six minutes from the end to put the Celts into the next round.
Stocksbridge Park Steels roared into the final qualifying round with a 5-3 triumph at Kidsgrove Athletic, while Shaw Lane AFC also stormed through with a 3-0 win at home to Whitby Town.
The draw for the third and final qualifying round takes place today.
In the EvoStik League Premier, Frickley Athletic moved up a place from 22nd to 21st with a 3-1 success at Ashton United. Frickley’s Kudakwashe Muskwe, Tyler Williams and Josephn Thornton all netted a goal each.
In First Division North, Tadcaster Albion climbed to ninth with a 5-2 win at third-bottom Ossett Albion.
Connor Sellars was at the double for Albion who also struck through Jason Myscoe, Thomas Corner and an Oliver Yates own goal. James Eyles bagged a brace for Albion, including one penalty.
Ossett Town are fifth after drawing 2-2 at Hyde United, with Jason Yates and Danny Frost both netting for the visitors.
Scarborough Athletic are up to second after seeing off visiting Droylsden 3-1 through a Carl Stewart double and Lamin Colley strike.
But Brighouse Town dropped to sixth after a 2-0 loss at Glossop North End, while second-bottom Goole AFC lost 4-0 at Bamber Bridge.