Shinji Okazaki’s header and instinctive strike put Leicester in control, but Gary Cahill nodded in before the break.
Cesar Azpilicueta equalised with a long-range effort before Foxes defender Marcin Wasilewski was sent off.
Fabregas’ clipped finish put Chelsea ahead early in the first extra period and a powerful shot sealed the win.
It had been a pulsating cup tie – particularly in the second half – with chances at both ends but Wasilewski’s dismissal for elbowing Diego Costa ultimately proved costly.
Chelsea deserved to progress but Foxes boss Claudio Ranieri will perhaps be buoyed by his side’s performance for much of the game, with glimpses of the incisive attacking and compact defending that helped them win the title last season.
The return of David Luiz has, so far, not solved Chelsea’s defensive troubles.
Chelsea have scored in every game this season and their attacking strength came to the fore towards the end of this game.
However, their defensive frailties should continue to be a cause for concern for boss Antonio Conte.
Only once in their previous 14 games in all competitions have the Blues kept a clean sheet – and they were needlessly exposed in the first half of this game.
Azpilicueta’s awkward touch when trying to deal with Ahmed Musa’s cross allowed Okazaki to nip in and flick a header over Asmir Begovic, and the Foxes forward then profited from a static backline to score a second.
“Chelsea’s decision-making when under pressure was poor,” said BBC Radio 5 live summariser and former Leicester defender Gerry Taggart.
“We are talking about basics. Clear the ball first, then ask questions. Some of Chelsea’s decision-making was abysmal.”
Fabregas has been unable to establish himself under Conte, with the Spanish midfielder restricted to appearances from the bench in the Premier League so far this season.
His only two starts for the Italian have come in the EFL Cup – but he was arguably the standout player for the Blues in this game.
The 29-year-old has been criticised for a lack of goals in the past – he scored six last season and five the campaign before – but took both his finishes well against the Foxes and also provided the assist for Cahill’s goal.
His was a performance to give Conte food for thought.