This SW6 derby was a close contest with chances aplenty yet few converted, but it was Chelsea who would emerge victorious over their rivals Fulham thanks to a volley from Mason Mount in the second-half.
Both sides spurned a number of opportunities in the first-half, which ended with the dismissal of Cottagers defender Antonee Robinson for a cynical challenge in Cesar Azpilicueta.
Despite holding strong for the majority of play after the break, it always seemed merely a matter of time before Chelsea would find the breakthrough, and Mount made no mistake with his chance in the 78th minute to hand his side a crucial victory.
The result leaves Fulham 18th in the Premier League table, now four points from safety, while the Blues climb to 7th.
Story of the game
Meanwhile, Frank Lampard made three changes, one of which was a necessity as the suspended N'Golo Kante dropped out for Jorginho in midfield, while Olivier Giroud and Antonio Rudiger came in for Timo Werner and Kurt Zouma in attack and defence respectively.
A minute of silence was held before the match in memory of Fulham icon Trevor "Tosh" Chamberlain who sadly passed away last Sunday.
An early corner for Chelsea was drilled in low by Hakim Ziyech, searching for Mount on the edge of the area, but the England international's shot was blocked and cleared well by the Fulham defence.
The decision to start Giroud nearly paid dividends for Lampard early on. A lovely series of interchanges between him and Mount left the home back line at sixes and sevens and eventually led to the Frenchman unleashing a fierce strike towards the far post; it needed to be tapped wide by his outstretched compatriot Alphonse Areola.
Following something of a shaky start - at one point, centre-back and captain for the day Joachim Andersen felt obliged to bellow, "more guys, come on!" - the Cottagers found their footing and put together some really intricate sequences.
They could have been rewarded with a goal just past the 15-minute mark, but for some Herculean defending by their visitors. Antonee Robinson laid the ball off for Andre-Frank Anguissa in the box, but the shot was blocked by Thiago Silva and Ivan Cavaleiro's follow-up by Cesar Azpilicueta.
Moments later, a quick counter-attack saw Ademola Lookman maraud down the left flank. He drew Azpilicueta into a challenge and immediately fizzed the ball into the feet of Cavaleiro, who turned on the spot but struggled to bring it under his control and the chance petered out.
Chelsea had emerged unscathed thus far from a wave of Fulham attacks and wanted to swing the game back into their favour. Ziyech, lively in the opening exchanges, took aim from roughly 20 yards out, but Areola got down well to save and held onto the ball with Giroud lurking for the rebound.
They so nearly took the lead with 23 minutes gone. A dangerous cross from Ben Chilwell met the head of Robinson, who only managed to clear as far as the feet of Mount in the box. The 22-year old brought the ball down and shot in one sweeping motion, leaving Areola stranded, but it rattled against the crossbar and Fulham survived.
Another golden opportunity came from a corner in the next phase of play. Ziyech's vicious delivery was glanced towards goal by Antonio Rudiger, but the German's effort was saved magnificently by Areola who palmed the ball away.
This was shaping up to be a strange match in the most entertaining of ways. Had it not been for their French shot-stopper, Fulham could quite easily have been down three goals within 25 minutes to their rivals, who were also lucky not to have conceded so far.
Just past the half-hour mark, Christian Pulisic latched onto a loose ball on the edge of the area and struck first-time. He failed to trouble Areola this time, but Fulham needed to cut down on surrendering these opportunities; it felt like just a matter of time before it would come to bite them.
But football's always been a game of surprises; it would have been just one of those to see the Cottagers take the lead on 40 minutes, and yet they should have done just that. A low cross from Kenny Tete found Cavaleiro just eight yards from goal, but he sent his effort too high with Edouard Mendy stranded.
One of the most eventful goalless first-halves you'll see enjoyed yet more drama just before the break when Robinson was dismissed for a cynical tackle on Azpilicueta. The contact was minimal, but the USA international was taking risks with a lunge like that and, in the eyes of referee Peter Bankes, he had to walk.
That was a big blow for Fulham who were very much in this game in the first-half; getting anything from this game now would require nothing short of a spectacular effort after the break.
With his side now a man down, Parker shuffled things slightly at the break but kept retained in the 10 he still had out on the pitch; Ola Aina shifted across to left-back and Lookman and Bobby Decordova-Reid were brought deeper to create a 4-4-1 formation with Cavaleiro leading the line.
To their credit, Fulham lost none of their energy and enthusiasm. With 52 minutes gone, Cavaleiro swung a cross-field pass out to Aina on the left. The Chelsea academy graduate wove onto his stronger right foot and let fly from the edge of the area, but he failed to trouble Mendy.
Chelsea had to be patient here; it was quickly apparent that their opponents weren't going to stop fighting to remain in this contest until their bodies could yield no more. For 10 straight minutes they probed and probed relentlessly, but were struggling to break down a resilient Fulham defence.
But Lampard sought the breakthrough for his side as soon as possible. In the 64th minute, he replaced Jorginho with forward Tammy Abraham, a change which saw his side line up in something resembling a 4-2-4.
Abraham nearly scored with his first touch when Chilwell crossed into the box, but his header was directed straight at Areola. In the next phase, Pulisic lunged a the back post for an Azpilicueta cross but the ball evaded his reach by a matter of inches.
This match was livening up as it edged into its final 20 minutes, and it was now Fulham's turn to threaten the opposition goal. A clearance from Mendy fell straight to Cavaleiro; with the goalkeeper of his line, the forward tried his luck, but Silva managed to get back and block the shot.
Playing 10 against 11 is an almighty task, one that's difficult to survive unscathed. Chelsea's breakthrough came 78 minutes in when a searching cross from Chilwell was palmed away by Areola, but only as far as Mount who lashed at goal on the volley with no one there to stop it.
Perhaps some of the visitors' urgency dwindled from that point, but they were desperate to hold onto their lead and were let off when substitute Aboubakar Kamara opened up space for a shot on the edge of the box, but his effort landed straight in the palms of Mendy.
The Blues had a gilt-edged chance to extend their lead deep into stoppage time. Pulisic carried the ball away from a Fulham corner and fed Timo Werner through on goal. The forward looked certain to score as he bore down on goal but skewed his resultant effort.
It would finish 1-0 to the visitors; Fulham's nearly 15-year winless run against their SW6 rivals continues.
Takeaways from the match
Fulham hold strong despite Robinson's dismissal, but energy is finite
The hosts here were going well in the first-half until they were reduced to 10 men, and didn't start the second badly at all either. But they were always going to run out of steam eventually, and may look back on one or two missed chances with regret.
It's been a damaging day for Fulham, losing here and seeing fellow relegation candidates Brighton & Hove Albion and West Bromwich Albion both pick up points elsewhere, but they still have games in hand to improve their points tally.
Chelsea grind to much-needed victory
One win in their last six Premier League games made this a must-win for Lampard's side, not just because of the bragging rights at stake. They were helped by the sending off, but quality also prevailed in the end, Fulham unable to withstand the physical and technical strain Chelsea placed on them.
It wasn't always pretty, but they did keep probing and were rewarded for their persistence in the end, and will be hoping this victory can set them on a run which sees them surge up the standings and back into Champions League contention.
Fulham: Areola; Aina (Bryan 82'), Andersen, Adarabioyo; Tete, Reed, Anguissa, Robinson; Decordova-Reid (Kamara 82'), Cavaleiro (Onomah 78'), Lookman.
Subs not used: Rodak, Odoi, Hector, Ream, Kebano.
Chelsea: Mendy; Azpilicueta, Rudiger, Silva, Chilwell; Kovacic, Jorginho (Abraham 64'), Mount; Ziyech (Hudson-Odoi 73'), Giroud (Werner 73'), Pulisic.
Subs not used: Arrizabalaga, James, Zouma, Emerson, Gilmour, Havertz.
Following defeat here, Fulham return to their homely Cottage on Wednesday evening, with high-flyers Manchester United their visitors this time around.
Meanwhile, Chelsea a play a day later when they make the trip to Leicester City on Thursday.