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.
Celtic, which overcame Barcelona 2-1 in last season's competition, created the better chances for much of the encounter, but the hosts were left to rue captain Scott Brown's dismissal, which came for kicking out at Neymar.
Even at the man disadvantage, Celtic looked set for a valuable point until Fabregas latched onto substitute Alexis Sanchez's cross from the right.
Barcelona arrived at Celtic Park without Lionel Messi but still boasted a strong starting lineup with Fabregas and Neymar both starting for Gerardo Martino’s men. Barcelona dictated possession in the early stages, but it was Celtic that created the game's first chance, with Georgios Samaras forcing a near-post save from Victor Valdes after being played through by Anthony Stokes.
Despite its dominance, it took Barca 16 minutes to craft its 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 Barca labored 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 Brown was sent off in the 59th minute for appearing to kick Neymar in the back after he brought down the Brazilian following his surging run into the Celtic half.
Barcelona unsurprisingly seized control once again from there as it looked to make its numerical advantage count, but 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 counterattack a few moments later as Fabregas met substitute Alexis' pinpoint 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.