But the Italian’s brief moment of pleasure will already have turned to anxious thoughts of how he can best patch up such a painfully leaky defence before Saturday’s high-profile Premier League match at London rivals Arsenal.
One of his tactics is likely to feature recalling experienced captain John Terry following injury.
Conte, like his Italian counterpart Claudio Ranieri, made seven changes to his starting line-up last night, but significantly made only one switch to his back four, handing expensive summer signing left-back Marcos Alonso a debut.
But the plan to instil some continuity into his defence backfired in dramatic fashion as Leicester hit two unanswered first-half goals courtesy of Japanese striker Shinji Okazaki.
It took an injury-time effort from Chelsea’s stand-in skipper Gary Cahill, badly at fault for Leicester’s first goal, to get the visitors back into the game seconds before the break.
And their equaliser, just four minutes after the restart, came from another man at the back looking to make amends in Cesar Azpilicueta.
But it was only after the introduction of Diego Costa as a second-half substitute and the subsequent sending-off for Leicester central defender Marcin Wasilewski that Chelsea really looked on top.
That red card, for a second bookable offence, left Leicester running on empty going into extra-time and Fabregas made them pay in style.
The Spanish midfielder, yet to start a league game under Conte, had been one of Chelsea’s better players throughout the tie and he came into his own when it really mattered.
He put the visitors in front for the first time on the night with a well-taken right-footed shot, following a deft touch from recently introduced substitute Eden Hazard, just over a minute into the first period of extra-time.
The former Arsenal and Barcelona player then rifled his second into the roof of goalkeeper Ron-Robert Zieler’s net from a half-cleared corner two minutes later.
There was no coming back for a weary Leicester side now with Saturday’s early kick-off at Manchester United no doubt coming into the tactics of Ranieri.
And it had all started so well for a Leicester side looking to inflict their first ever major cup competition defeat over the Chelsea side they replaced as Premier League champions last season.
Chaos in the Chelsea defence led to the 18th-minute opener as an Ahmed Musa cross bounced off a hesitant Azpilicueta; keeper Asmir Begovic and David Luiz were equally indecisive to react to the danger compared to the alert Okazaki who headed in from an acute angle.
Conte had planned to use this game to give fringe players a chance but also to try to forge more understanding between his back four after their less than convincing display in last Friday night’s home defeat by Liverpool.
No wonder the Italian coach looked increasingly frantic and agitated on the touchline. Not that the former Italy national manager hardly ever looks calm and satisfied.
More poor defending preceded Leicester’s second as an excellent Andy King pass played in Okazaki and he bounced his shot into the ground to beat Begovic at the same post where he scored his first.
First-half injury-time handed Chelsea a corner where Cahill lost his marker Wes Morgan and headed it back across goal and, not unlike the Leicester goals, the ball crept just over the line.
It was possibly more than Chelsea deserved for their first-half efforts and some of the Leicester players looked downhearted as they walked off to the half-time whistle.
They looked even more disconsolate less than four minutes after the restart when Chelsea were back on level terms. Fabregas crossed from the right and Wasilewski’s weak clearance was hammered back past him and into the net by Azpilicueta.
The Spanish full-back was a full 20 yards out and it was the sweetest of strikes from the right boot to mark only his fourth ever goal for the club.
With extra-time looming both managers sent on their big goalscoring guns with Costa making all the difference for Chelsea but Jamie Vardy failing to make such a big impact for Leicester.
At least the Midlands side, unlike Chelsea, now have the relative glory of saying they can concentrate on the Champions League.
Zieler 5; Simpson 6, Morgan 6, Wasilewski 5, Chilwell 6: Gray 6 (Amartey 90, 6), King 7, Drinkwater 6, Schlupp 6; Okazaki 8 (Ulloa 75, 6), Musa 7 (Vardy 76, 6). Subs: Hamer,, Kapustka, Ulloa, Mahrez, Benalouane.
Begovic 5: Azpilicueta 7, Cahill 6, David Luiz 5, Alonso 5; Fabregas 6, Matic 5; Moses 6, Loftus-Cheek 5 (Costa 67, 7), Pedro 5 (Hazard 89, ); Batshuayi 5 (Chalobah 80, 5). Subs: Courtois, Kante, Willian, Aina.
Bobby Madley 6/10