Swansea rallied late on in their search for a valuable equaliser, but Chelsea held on to boost their top-four hopes while leaving the hosts now just one point clear of danger at the bottom.
Swansea had seen the three teams below them in the table all pick up points earlier in the day, with West Bromwich Albion and Southampton both winning, while Stoke City earned a point against Liverpool in a result which would have also been of interest to Chelsea.
The hosts' day did not get any better in the opening stages of this match either as they found themselves behind after only four minutes when Fabregas swept a classy finish into the top corner to bring up his 50th Premier League goal - only the third Spaniard to reach that milestone.
Swansea responded relatively well to such an early setback without ever really looking like breaking through the Chelsea defence in the first half, and they were fortunate not to fall two behind midway through the half when Alfie Mawson lifted the ball over his own goalkeeper, skimming the top of the crossbar on its way over.
It was a first half lacking in clear chances, though, as Chelsea struggled to significantly build on their early goal despite some eye-catching link-up play between the likes of Eden Hazard, Fabregas and Olivier Giroud - the latter of whom was again preferred to Alvaro Morata up front.
Swansea did show more attacking threat in the early stages of the second half when Martin Olsson burst down the left flank and whipped a cross into the middle which Andre Ayew attacked well, but Antonio Rudiger did just enough to prevent a clean header.
The match soon fell back into a familiar toothless rhythm, though, and it wasn't until the final 20 minutes that either side really pushed for the second goal of the game.
A powerful run from Emerson Palmieri kickstarted the flurry of chances as he carried the ball from inside his own half to the edge of the area before drilling a low strike towards goal which Lukasz Fabianski got down to save.
Swansea responded with a chance of their own moments later as Chelsea made a mess of defending a corner inside the box, allowing Jordan Ayew to loop a head narrowly off target with Thibaut Courtois out of position.
The visitors were back on the front foot just one minute later as Hazard and Fabregas linked up well inside the box, with the latter's effort hitting Victor Moses before trickling agonisingly wide of the target from close range.
It was Swansea who applied all of the late pressure in the closing 15 minutes, though, and arguably their best chance came when Andre Ayew collected a pass from Tom Carroll before creating space to shoot and curling his finish narrowly wide of the far post.
Kyle Naughton then drew a save from Courtois with a well-struck effort from the edge of the area before Hazard missed the chance to put the game to bed when he fluffed his lines from the edge of the area after a clever knockdown from Giroud.
Swansea were soon back on the front foot, with Courtois doing well in difficult conditions to deal with an Alfie Mawson strike before Carroll curled another long-range effort narrowly wide of the target.
There was still time for one last chance in the final minute of the 90 as the ball fell to Wayne Routledge inside the area, but his finish was tame as Courtois again collected it to ultimately end Swansea's hopes of a crucial comeback.
The result - Conte's 50th win in the Premier League - sees Chelsea move to within striking distance of the top four until at least when Tottenham Hotspur face Watford on Monday night, also setting up an important showdown with Liverpool next weekend.
Swansea, meanwhile, are now winless in six Premier League games to cap off a miserable day for the South Wales outfit, with the defeat also ending their five-match unbeaten streak at home in the league.
SWANSEA (3-4-3): Fabianski; Naughton, Van der Hoorn, Mawson; Roberts (Carroll 63'), Ki, King (Dyer 58'), Olsson (Routledge 81'); A Ayew, J Ayew, Clucas
CHELSEA (3-5-1-1): Courtois; Azpilicueta, Cahill, Rudiger; Moses, Fabregas (Pedro 81'), Kante, Bakayoko, Emerson; Hazard (Willian 81'); Giroud (Morata 85')