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.
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