A sixth Premier League victory in a row on home soil moves Chelsea to within a point of second-placed Manchester United, who were held to a draw at home to Burnley on Tuesday afternoon.
Tomer Hemed was wasteful with a headed chance at the other end, though, before Chelsea began to turn the screw in search of an opener.
Shots from Victor Moses and Fabregas were also kept out by Ryan, with an organised Brighton side doing what was required to get themselves into the interval on level terms.
That changed soon into the second half, however, as Cesar Azpilicueta assisted Morata for the sixth time this season just 51 seconds after the restart, sending a bullet header past Ryan.
After Hazard and Alonso came close to doubling the hosts' tally, the latter finally got the better of Ryan on the hour with a deft header to seal the points.
Brighton, with just one win in this fixture in eight attempts going back to 1933, were given the chance to attack a little more from that point on.
The Seagulls were left open to counter-attacks, though, which in-form forward Hazard almost capitalised on twice in the final 25 minutes.
Shane Duffy glanced the post from the final opportunity of the match after being spotted in space 13 yards from goal, as Chelsea successfully saw things through for three more points.
Chelsea: Courtois; Azpilicueta, Cahill, Rudiger; Moses, Kante (Drinkwater 85'), Fabregas, Bakayoko, Alonso; Hazard (Willian 73'), Morata (Batshuayi 82')
Brighton & Hove Albion: Ryan; Schelotto, Duffy, Dunk, Suttner; Stephens, Propper; Kayal (Gross 73'), March (Knockaert 82'), Izquierdo; Hemed (Murray 80')