Marco Silva's troops gave a good account of themselves in the capital and subjected the home crowd to a few nervy moments, but Cole Palmer's ninth top-flight goal of the season from the spot was enough for the hosts to come out on the right end of the scoreline.
A triple injury boost for the Blues saw all of Ben Chilwell, Benoit Badiashile and Carney Chukwuemeka passed fit to make the substitutes bench, while Fulham wide man Willian held his spot in the Cottagers' XI against his former employers.
As expected, Mauricio Pochettino's troops saw most of the ball in the earliest exchanges, but as was the case in the first half of their EFL Cup clash with Liverpool, Fulham's defence was holding firm and denying the hosts any clear openings.
Not until the 15th minute did the hosts have a chance of note, and even then Conor Gallagher's tame shot did not trouble Bernd Leno as it sailed over the top, although the German was left rooted to the spot as Armando Broja met an Enzo Fernandez delivery in the 21st minute, only to send his header wide.
Down the other end of the pitch, Dorde Petrovic was forced into action twice in the space of a matter of moments to keep Fulham out, firstly saving comfortably from Andreas Pereira in the 26th minute before denying Harry Wilson from close range after the Welshman caught Levi Colwill napping at the back stick.
An otherwise tepid first half was seemingly destined to end goalless, but deep into injury time, Raheem Sterling checked back inside the Fulham box and was tripped by Issa Diop's trailing foot, leaving Anthony Taylor with no choice but to point to the spot.
Despite the best efforts of the Fulham faithful in front of him, Palmer comfortably sent Leno the wrong way to give Chelsea the lead at the end of a first half in which officiating controversy also reared its ugly head again; Malo Gusto got away with just a yellow card for a crunching challenge on Willian in the 39th minute.
Both sides' first-half showings left a lot to be desired, but the tempo certainly kicked up a notch at the start of the second 45, as both Palmer and Raul Jimenez sent efforts over the top within 10 minutes of the restart.
For all of Fulham's belief, they were doing little to worry Petrovic in the Chelsea goal until the 73rd minute, when Jimenez met a Pereira cutback but failed to get enough venom on his strike to beat the Serbian, who palmed the ball clear down to his right.
In a further attempt to shut up shop, Pochettino brought on a fit-again Chilwell for his first appearance since the end of September, while Noni Madueke was also introduced to provide an injection of ingenuity on the right.
With 83 minutes gone, the latter sent a deflected cutback into the Fulham box, which fell at the feet of Gallagher, but the Englishman's clever outside-of-the-foot effort clipped the post to keep Fulham's hopes of a late fightback alive.
Willian, who else, was presented with a golden chance to haunt his former club with an 86th-minute free kick from just outside the box, but the Brazilian's powerful set piece curled just over the bar, and Chelsea emerged unharmed from a couple more scrambles to put their rivals to the sword.
Thanks to a third top-flight win on the trot, Pochettino's side have risen above Newcastle United and Manchester United into eighth place in the table, while Fulham stay put in 13th spot.
EFL Cup semi-final seconds legs are next on the agenda for both sides, as Chelsea now have a 10-day break before their meeting with Middlesbrough on January 23 - where they will attempt to overturn a 1-0 deficit - one day before Fulham also bid to bounce back from their 2-1 loss to Liverpool.