The Gunners were knocked out 4-3 on aggregate by Barcelona after losing the last 16 second leg 3-1 at the Nou Camp on Tuesday night, with Barca's first goal coming directly from a mistake by the Spanish international.
Fabregas attempted to find Jack Wilshere with a back-heel just outside his own penalty area, but the pass was intercepted by Andres Iniesta. The Barca star jinked past three Arsenal defenders before delicately chipping a pass into the area where Lionel Messi lifted the ball over the on-rushing Manuel Almunia.
Arsenal were handed a lifeline just after half-time as Sergio Busquets headed a corner into his own net, but a flowing move in the 69th minute ended with a Xavi goal which was followed by a Lionel Messi penalty just two minutes later.
Arsenal felt aggrieved in the 56th minute as Robin van Persie picked up a second yellow card for a shot on goal after the referee had blown for offside, despite the Dutchman's insistance that he couldn't hear the whistle over the 95,00 fans in the stadium.
Fabregas however, doesn't believe the loss is down to Van Persie's dismissal, but his own mistake.
Speaking on Twitter, Fabregas said: "I take full blame for the result tonight. One of the worst moments of my life. I apologise."
He added: "Great support from the Arsenal fans."
However, the former Barcelona youth player was pleased with his side's effort, believing they did all they could do, whilst also adding that he played through the pain barrier at the Nou Camp, something he wasn't able to do last season.
"It's a real pity we've gone out tonight. We caused them a few scary moments, especially when we scored," he told Uefa's official website after the match.
"But that's football and it wasn't our night. I'm proud of the team; we stood up to them and defended well when we were supposed to.
"Personally, I enjoyed the first 15 minutes but then felt a slight pain in the back of my leg and carried on for as long as I could. I wish Barcelona the best of luck in the tournament and hope they can get as far as possible."