The result was a little harsh on Brighton, who responded well to their nightmare start but failed to score for the ninth time in their last 12 league matches.
Brighton could not cope with Hazard throughout the match and paid for giving the Belgian too much space in the box for the opening goal on three minutes.
Last season's PFA Player of the Year was involved again when Willian rapidly doubled Chelsea's lead from Michy Batshuayi's flick at the end of a flowing move.
There was little that Ryan could do to stop Willian's 20-yard thunderbolt, but he did save well from the Brazilian, Batshuayi and Tiemoue Bakayoko as Chelsea cut open the Seagulls at will in the first half.
However, Chelsea's attacking excellence was only half of the story as Brighton could have had two penalties before the break.
Ezequiel Schelotto was tripped by Willy Caballero, and later felled by Bakayoko, but referee Jonathan Moss, who was unsighted for the first and stronger appeal, turned down both shouts from the Seagulls.
Fortune continued to escape the hosts in the second half as Davy Propper headed against the crossbar, before Chelsea hit the woodwork themselves as Ryan tipped Willian's curling free kick onto the post.
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.
Hazard got a deserved second goal on 77 minutes when he calmly slotted into the bottom corner from 15 yards, after being invited to run into the box by a terrified Brighton defence which continued to back off.
To cap things off for Chelsea, Moses scored his first league goal in 14 months by controlling a perfectly-weighted lofted pass by substitute Charly Musonda, and poking the ball past Ryan one minute from time.
A first win in six in all competitions lifts Chelsea to third in the table, while Brighton remain 16th and could be sucked further into relegation trouble by the end of the weekend's fixtures.
BRIGHTON (3-5-2): Ryan; Schelotto, Duffy, Goldson, Dunk, Suttner; Propper, Stephens, March (Izquierdo, 83'); Gross (Kayal, 68'); Hemed (Murray, 68')
CHELSEA (3-5-2): Caballero; Azpilicueta, Christensen (David Luiz, 58'), Rudiger; Moses, Bakayoko, Kante, Alonso (Zappacosta, 75'); Willian (Musonda, 81'), Batshuayi, Hazard