Neil Lennon's men slipped to Champions League defeat thanks to a late goal from the former Arsenal captain after Scott Brown had been sent off for kicking out at Neymar
Fabregas found the net 14 minutes from time in the Group H encounter as the Liga champions took advantage after Scott Brown had been sent off in controversial circumstances just before the hour mark.
Celtic - who overcame Barcelona 2-1 in last season's competition - created the better chances for much of the encounter, but they were left to rue Brown's dismissal, which came following a clash with Neymar.
They were ultimately made to pay as former Arsenal man Fabregas struck to end their hopes of a repeat of last year's heroics.
Celtic made just one change to the side that beat Kilmarnock 5-2 at the weekend, midfielder Mikael Lustig coming in for Teemu Pukki.
The visitors, meanwhile, had to cope without talisman Lionel Messi, who missed out due to a hamstring injury, with Neymar replacing him at the head of the attack. Xavi came back into the side in place of Alexis Sanchez.
Barcelona dictated possession in the early stages, but it was Celtic who created the game's first chance, Georgios Samaras forcing a near-post save from Victor Valdes after being played through by Anthony Stokes.
Despite their dominance, it took Barca 16 minutes to craft their first opportunity as Neymar fired wide from range.
Andres Iniesta and Adriano then both saw chances go begging, although Celtic goalkeeper Fraser Forster still had little to do in the first half as Gerardo Martino's men laboured to try and find a way through.
However, the Catalan club went close to taking the lead five minutes before the break when Neymar turned Dani Alves’ right-wing cross narrowly wide of the post.
The hosts made the better start to the second half as they pushed to try and take an unlikely lead, but their hopes were dealt a huge blow when midfielder Brown was sent off in the 59th minute for appearing to kick Neymar in the back after the 28-year-old had brought down the Brazilian following his surging run into the Celtic half.
Barcelona unsurprisingly seized control once again from there as they looked to make their numerical advantage count.
Still, Celtic defended resolutely, with Barca reduced to speculative efforts from Marc Bartra and Fabregas.
Neil Lennon's men did look dangerous on the break despite Brown's dismissal and James Forrest forced a fine diving save from Valdes after 73 minutes before Charlie Mulgrew headed wide from the resulting corner.
But the hosts’ strong resistance was finally broken by a superb Barcelona counter-attack a few moments later as Fabregas met substitute Alexis' pin-point cross with an excellent header that looped beyond Forster and into the net.
Forster produced a fine double save to deny Neymar and Alexis in quick succession, before again preventing Neymar from adding a second, as one proved to be enough for Martino's men.