Celtic captain Scott Brown was sent off just before the hour mark for kicking out at Neymar and Barcelona make their numerical advantage count with Fabregas scoring the only goal of the game 14 minutes from time.
The defeat is Celtic’s second in Group H and leaves them facing an uphill battle to reach the knockout stages, while Barcelona made it two wins out of two to top the group.
Barcelona, as expected, dominated possession from the first whistle, but it was Celtic who created the first chance of the game after five minutes.
Kris Commons found Anthony Stokes in space and he slipped in Georgios Samaras inside the box with a clever pass, but Barcelona goalkeeper Victor Valdes was quickly off his line to keep out his effort from a tight angle.
Barcelona continued to push and probe in search of an opening, but they were struggling to make any inroads into a well-organised Celtic rearguard.
Adriano had a sight of goal on 26 minutes when Neymar found him down the left, but under pressure from Efe Ambrose he could only drill his shot into the side-netting.
Barcelona created their best opening of the first half four minutes before the break when Alves flashed over a cross from the right and Neymar got ahead of his marker Virgil van Dijk to flick his shot from 12 yards just wide of the target.
Celtic suffered a major blow just before the hour mark when Brown was shown a straight red card after appearing to kick out at Neymar after bringing down the Brazilian in full flow.
Despite being down to 10 men Celtic went close to breaking the deadlock on 74 minutes when the ball dropped to substitute James Forrest inside the box and he fired in a fierce half-volley which Valdes brilliantly tipped over the crossbar.
From the resulting corner Celtic missed the best chance of the game as Valdes came out for the cross and got nowhere near it, but Charlie Mulgrew could only head his effort wide of the target with the goal at his mercy.
Celtic were made to pay for that miss as just two minutes later Barcelona grabbed the lead with substitute Alexis Sanchez crossing for Fabregas to head home from six yards.
Forster prevented Barcelona from adding a second goal with a superb double save two minutes from time to deny Neymar and then Alexis as Barcelona looked to finish off the game.
Neymar had the final chance of the game in stoppage time when Alexis sent him clear on goal only for Forster to produce another great save to keep out the Brazilian, but it did not matter as Barcelona took all three points.