Palmer sinks Fulham to lift Chelsea gloom
Palmer converted a penalty late in the first half at Stamford Bridge to erase the bitter taste from Chelsea's shock 1-0 loss against second-tier Middlesbrough in the semi-final first leg on Tuesday.
That League Cup calamity was just the latest disappointment in a troubled season for Mauricio Pochettino's side.
Chelsea were hardly much more dynamic against Fulham, but Pochettino will take heart from the way they ground out a third successive Premier League victory.
After a 10-day winter break, the Blues will have a chance to overturn the League Cup deficit when they return to action against Middlesbrough in the second leg on January 23.
Chelsea are still a work in progress in Pochettino's first season, with his decision to go with a largely youthful team inevitably producing highs and lows.
In the circumstances, Fulham were ideal opponents for Chelsea to lift the gloom, given they are without a win at the Bridge since 1979.
The clubs are separated by just two miles in west London, but they have traditionally been light years apart in terms of success on the pitch.
Chelsea's recent struggles had allowed Fulham to dream of a rare victory on enemy turf and there was audible frustration among Blues fans as soon as passes started to go astray from their spluttering team.
That angst transmitted to the pitch as Chelsea laboured to establish any rhythm, with Conor Gallagher's wayward strike from distance epitomising their struggles.
Palmer keeps cool
Nicolas Jackson's absence on Senegal duty at the Africa Cup of Nations and Christopher Nkunku's latest injury have diminished Pochettino's attacking options.
Armando Broja was deputising for Jackson, but he wasted a good chance when he headed wide from Enzo Fernandez's cross.
Looking to capitalise on Chelsea's woes, Fulham launched an incisive break that ended with Harry Wilson's close-range effort forcing a good save from Djordje Petrovic.
When another Chelsea attack petered out, Fulham fans mocked the hosts' expensively assembled side with chants of "what a waste of money".
Despite their lethargic display, Chelsea rallied to take the lead in first-half stoppage time.
Issa Diop conceded a penalty with a rash challenge on Raheem Sterling and Palmer stepped up to smash the spot-kick past Bernd Leno.
It was the 21-year-old's nerveless fifth successful penalty for Chelsea showed why team-mates have nicknamed him 'Cold Palmer'.
With nine goals since signing from Manchester City in September, Palmer has been one of the few bright spots in Chelsea's troubled campaign.
Driving forward with intent, Palmer curled just wide early in the second half.
Chelsea lost their way again as the half wore on and Fulham striker Raul Jimenez headed over before testing Petrovic with a dangerous low strike.
Gallagher hit the woodwork with a clever effort using the outside of his foot, but Fulham finished strongly and only some last-ditch defending stopped the Blues suffering fresh indignity.
With Chelsea just about holding on, there was good news in the closing minutes when England left-back Ben Chilwell made his return from four months out with a hamstring injury.