Gary Rowett's Rams produced a dogged display of defensive solidity in the first leg on home soil in order to take what could be a crucial lead to Craven Cottage in the return fixture next week.
Fulham enjoyed much of the early possession but were left frustrated by a compact home defence, soon leading Derby to establish a counter-attack that saw Jerome almost scamper clean through on goal, but for a last-ditch interception from away stopper Marcus Bettinelli.
County's defensive pragmatism stifled the Cottagers for much of the first period, although Matt Targett created a moment of panic inside the County box, his drilled cross flying through bodies before falling kindly into the grasp of Scott Carson.
Derby offered glimmers of a threat on the counter and they took the lead in 34th minute with their first clear-cut chance. Craig Forsyth broke expertly down the left channel to send a curling ball into the middle, where Jerome towered above Targett to power a header beyond Bettinelli.
It took the best part of 38 minutes for a tepid Fulham to register a shot on target, Aleksandar Mitrovic holding the ball up well before turning to fire a shot straight at a grateful Carson in the centre of goal.
Fulham were much improved after the break and began to bang on the door with more force, Kevin McDonald bending a wonderful effort beyond Carson only to see his shot cannon back off the crossbar.
Back the visitors came once more with County penned inside their own half, this time Stefan Johansen finding Ayite off the shoulder of Curtis Davies, only for the forward to blaze his first-time volley over the target.
With Fulham pushing for a leveller, the Rams sought to profit on the break and they almost grabbed a surprise second goal when Tom Lawrence charged away at pace, although his flashing cross into the box evaded substitute Ikechi Anya by the finest of margins.
Slavisa Jokanovic's charges continued to toil on a perplexing evening for the away side and on another night Tom Cairney might have come away a hero, but the youngster saw his curling shot from range flash just inches wide of the post with Carson beaten.
Having scored 22 goals in the final 15 minutes of games, Fulham went into the dying embers looking for an equaliser to take back to London, but they were continually thwarted by a resilient Derby side who will now fancy their chances of ousting the favourites at the first hurdle come next week.
DERBY COUNTY (3-4-3): Carson; Keogh, Davies, Forsyth; Wisdom, Huddlestone, Johnson, Weimann (Hanson 90'); Vydra (Anya 68'), Lawrence, Jerome (Nugent 68')
FULHAM (4-3-3): Bettinelli; Fredericks, Odoi, Ream, Targett; McDonald, Johansen (Norwood 88'), Cairney; Ayite (Piazon 76'), Mitrovic, Sessegnon (Kebano 76')