Brighton's final opportunity to get back into the game was spurned on 66 minutes when Caballero blocked from Schelotto at point-blank range, after the Brighton wing-back's persistence earned him the chance.
That said, they have only been victorious in one of their last five away top-flight matches, and the goals have dried up in recent weeks due to the struggles of Alvaro Morata.
After that early double blow the Seagulls grew into the game and felt they should have been awarded a penalty around 15 minutes in as Schelotto looked to have been brought down in the box by Caballero, however referee John Moss waved away their pleas. The 89th minute strike by Moses capped the Blues party after getting an assist from Musonda.
His back-heel to set up Willian for Chelsea's second was timed to perfection, highlighting his vision and impressive link-up play.
Chelsea are in the midst of a run of three consecutive scoreless games for the first time in their history but they are hoping to put that behind them as they battle for a top-four place in the Premier League.
Conte's side were without a win in five games in all competitions heading into the clash on the south coast, but Hazard set the Blues on their way in the third minute before Willian rounded off a sensational team goal three minutes later to give the visitors the flawless start.More news: House Ethics Committee member used taxpayer money to settle sexual misconduct complaint
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Chelsea team: Caballero; Azpilicueta (c), Christensen, Rudiger; Moses, Bakayoko, Kante, Alonso; Willian, Batshuayi, Hazard.
The two sides headed into half-time with Chelsea taking the lead.
Second-placed United eked out a 1-0 win over Burnley, Anthony Martial with the goal, and moved to within nine points of leaders Manchester City, who face Newcastle in the late game.
But they were profligate and eventually punished by Hazard, who netted a brilliant third 13 minutes from time. However, the Blues would not care about this as they had only managed to score one goal in four previous games in all competitions.
'But then after the first-half we had a problem.
The Belgian was played in by Willian, and with defenders on the back-foot, he drove into the area and finished coolly past Ryan in front of the Chelsea supporters.