Swansea's loss leaves them just a point outside the drop-zone following Southampton's win in the south coast derby, while Chelsea are now just two points off Tottenham in fourth, as Conte's men continue to fight for a Champions League spot.
As the Spaniard gave Chelsea the lead inside three minutes, Swansea failed to retaliate, and the chances continued to fall to those in blue.
"Sometimes when you are in a dip of form and you are struggling to get points, to win whatever it takes, to do it is always positive".
"When I arrived we were five points to the teams above, we depended on others to try and get out of this position", he said.
Premier League leading scorer Mohamed Salah had a rare off day as Liverpool stumbled in its chase for a top four finish by drawing 0-0 with Stoke on Saturday. The draw did little to help Stoke's survival bid as it remains in the relegation zone surrounded by teams with games in hand.
The Swans started to grow into the game towards the end of the half, but were unable to penetrate the Chelsea backline and test Thibaut Courtois.
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Chelsea are now the 5/4 favourites with 888sport to win the game and blow the top four race wide open while you can get odds of 2/1 on Liverpool taking home the three points and wrapping-up their Champions League place. Chelsea nearly doubled their lead through an unlikely source as Swans defender Alfie Mawson miscued an attempted clearance towards his own goal.
Andy King's heavy touch gifted Chelsea possession and he was duly punished with Fabregas sweeping home from 20 yards for his 50th Premier League goal.
With the final two games at the Liberty where the home fans remain noisily supportive, he could see nothing to worry about. Marc Albrighton was sent off for bringing down Zaha.
West Brom is five points from safety with two games remaining.
Cesc Fabregas scored the only goal in the 4th minute in a counter attack.
Intriguingly, Chelsea and Liverpool meet at Stamford Bridge next weekend, when the pressure could really be applied on Jürgen Klopp's side. Thanks to Dusan Tadic's brace (WATCH HERE) at St. Mary's Stadium, and Swansea's narrow defeat to Chelsea, Southampton now occupy 18th place and trail the Swans by a single point.