Without the injured Lionel Messi, Barcelona dominated possession but created few chances until the home captain Brown was handed a straight red card around the hour mark after clipping Neymar with a late tackle before appearing to kick the hand of the Brazilian as he raced past him.
The referee firstly produced the yellow card before turning to red with Neymar rolling around like he had been shot.
It left Celtic with a mountain to climb, and Fabregas secured the three points for his side when he nodded Alexis Sanchez's cross into the home net with a wonderful reverse header on 76 minutes.
Celtic - led superbly well by the rampaging Georgios Samaras - should have gone ahead seconds before Barca struck when Victor Valdes made a brilliant stop from a James Forrest volley before Charlie Mulgrew somehow headed wide of goal when he looked certain to score.
Celtic huffed and puffed despite the numerical disadvantage, but needed goalkeeper Fraser Forster to keep them alive with three world-class stops late on that denied Neymar twice and another effort from Sanchez that looked destined to hit the net.
Despite performing strongly, Celtic have lost their opening two matches to Milan and Barca meaning their next match against Ajax on October 22 is almost a must-win situation for the Scottish champions if they are to keep alive hopes of reaching the last 16 for a second straight season.
MAN OF THE MATCH
Georgios Samaras (Celtic): Lacked a bit of finesse, but never stopped working as Celtic were unfortunate not to emerge with a point from their evening's efforts.
40' - NEYMAR VOLLEY: Neymar clips a volley a couple of yards wide from the Dani Alves cross. Brilliant delivery and a fine effort.
43' - XAVI FREE-KICK: Xavi with the delivery. Pique can't get on the end of it, but Forster somehow scrambles it clear.
60' - BROWN SEES RED: Brown is off the park for a late challenge on Neymar. A horrid decision by the referee. Was surely only worth the yellow card. The match official gets that all wrong.
72' - VALDES HEROICS: Mulgrew with the long throw-in. Forrest smashes the volley at the goal, but Valdes with a magnificent tip over his bar. How did he keep that out?
73' - MULGREW MISSES GAPING GOAL: Mulgrew with a free header at goal. A couple of yards wide of goal. What a chance that is. Not taken by Celtic.
76' - FABREGAS WINNER: Fabregas secures the three points for his side when he nods Alexis Sanchez's cross into the home net with a wonderful reverse header on 76 minutes.
86' - FORSTER BRILLIANCE: The goalkeeper stops the Neymar shot before getting back to his feet to keep out Alexis.
88' - MORE FORSTER RESISTANCE: Another diving stop from Forster to deny Neymar. The English goalkeeper showing his class again at this level.
TALKING POINT: Did Scott Brown deserve to see red? By the letter of the law yes, but by sporting fair play no. No doubt Neymar's theatrical reaction prompted the referee to produce the red card. Changed the course of the match in Barca's favour with Celtic goalkeeper Fraser Forster having little to do for the first 60 minutes. Neymar should be embarrassed by his antics.
Barcelona manager Gerardo Martino has now guided Barcelona to nine wins in nine matches this season.
Barcelona have enjoyed 80% (or more) possession three times in the Champions League since the start of last season, all against Celtic.
Celtic: Fraser Forster 9, Adam Matthews 8, Emilio Izaguirre 6, Efe Ambrose 7, Virgil van Dijk 7, Charlie Mulgrew 6, Mikael Lustig 6, Brown 5, Commons 6, Samaras 9, Stokes 7..subs: Kayal 6, Pukki 6, Forrest 7
Barcelona: Valdés 8, Piqué 7, Bartra 7, Adriano 7, Alves 7, Fàbregas 8, Hernández 7, Iniesta 7, Busquets 7, Rodríguez 7, Neymar 7..subs: Song 7, Sánchez 7, Tello 7
THE MATCH IN A TWEET