Chelsea were 2-0 up inside six minutes on Saturday through Hazard and a magnificent strike from Willian - a one-touch move featuring backheels from Hazard and Michy Batshuayi illuminating the Amex Stadium.
Brighton, to their credit, did not buckle as Ezequiel Schelotto had a pair of penalty appeals turned away and Davy Propper hit the post.
But the majestic Hazard and Victor Moses put the result beyond doubt during the final quarter of an hour, ending a frustrating period for Conte's men.
They drew four consecutive matches in all competitions before Wednesday's penalty shootout triumph over Norwich City in the FA Cup – a victory that came at a cost as red cards to Alvaro Morata and Pedro left what had been a shot-shy attack since the turn of the year even more depleted.
Thibaut Courtois also missed out against Brighton, where Conte felt his team's performance scotched any talk of a crisis.
"Whoever said this has to pay attention to the stats. We're unbeaten in 11 games," he told a post-match news conference.
"It's normal that the press try to see the negative aspect and don't see the positives. But this team, with many problems, are unbeaten in 11 games in every competition.
"This means we are performing important work with these players to win in this way, despite five players less: [Cesc] Fabregas, [Danny] Drinkwater, Courtois, two suspensions in Pedro and Morata.
"Despite this, we won 4-0. That means that every player deserves his chance, and every player has to fight to play.
"Now we have to continue in this way. I'm pleased with the responses of all the players. To grow in this aspect is very important for us."
A class act. pic.twitter.com/WjOzYHJkJv— Chelsea FC (@ChelseaFC) January 20, 2018
Hazard dazzled throughout, scoring his first goals from open play since the start of December and Conte called for the Belgium star to maintain that level of performance in Wednesday's Carabao Cup showdown against Arsenal at Stamford Bridge, with the semi-final tie locked at 0-0 after the first leg.
"But, from tomorrow, every single player that scored today has to forget he scored and must try and score again against Arsenal.
"This is the most important thing: to be happy for the performance but, from tomorrow, to think about another important game against Arsenal, another important situation to try to help the team to win."
Andreas Christensen is expected to join his team-mates in training, with Conte reporting the head injury that forced the defender off after an hour was not as serious as feared.
"It was a simple contusion," he added. "The player is okay. From tomorrow's training session, he's ready."