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 strike before Foxes defender Marcin Wasilewski was sent off.
Fabregas' clipped finish put Chelsea ahead early in the first extra period and his powerful strike 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 the win in the end but Claudio Ranieri will perhaps be buoyed by his side's performance for much of the game, with glimpses of their incisive attacking and compact defending that helped them win the title last season on show.
