Fabregas' fourth-minute curler - his first goal in the league since August - was enough to see off Swansea, who remained competitive but without a cutting edge.
It was a game of two halves but the majority of the game proved to be stale and boring as both the teams did not possess any real goal threat.
The win moves Chelsea to within two points of fourth-placed Tottenham, who play Watford on Monday, but nine Premier League defeats this season have left Conte's men needing results elsewhere to go their way in the final weeks to avoid another campaign without Champions League football.
The battle is heating up at opposite ends of the table in the quest for Champions League qualification as well as top-flight survival. Today has been very positive to put that pressure on Tottenham.
"We started the game very well and dominated the game in the first half".
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"We were very good defensively and it's important for us to take three points with a good win".
"We played against a strong team", he told BT Sport.
"When I arrived we were five points off at the bottom but we've progressed and fight to stay in the division".
Despite trailing by one goal, the hosts grew in confidence as the game progressed, and they were able to pull some passes together, but their efforts to ensure they level up as Connor Roberts header assisted by Martin Olsson was cleared by the Blues defence.
Antonio Conte has highlighted Alvaro Morata's lack of goals as the main reason Chelsea have failed to retain their Premier League crown. "We tried to move the ball quicker in the second half", Carvalhal said.
"I think the players and I, we are trying to do our best".