The nine-time winners initially had no answer to the Cottagers' defensive supremacy and fell behind to a Willian opener, but inspired substitutions from Jurgen Klopp turned the tie on its head in the second 45, where Curtis Jones and Cody Gakpo hit two quickfire strikes to give Liverpool a slender advantage heading into the second leg.
The headline news from the Liverpool lineup was the expected promotion of Conor Bradley to the first XI in place of the injured Trent Alexander-Arnold, while Virgil van Dijk returned from illness to take his place at the heart of the backline.
Klopp's outfit, as anticipated, penned the Fulham defence into their own half straight away and had a sight of goal in the sixth minute of the contest, although Jones's strike from outside the box was meat and drink for Bernd Leno.
Fulham were evidently happy to absorb pressure and try to capitalise on any uncharacteristic Liverpool mistakes, and they did exactly that in the 19th minute, where Van Dijk was the guilty party on the edge of his own penalty area.
The Dutchman's misplaced header fell at the feet of Andreas Pereira, who cut back for compatriot Willian, and the Brazilian skipped past a tumbling Bradley before firing in at Caoimhin Kelleher's near post through Van Dijk's legs.
The Cottagers' rigid defensive structure and Liverpool's passive attacking play meant that the visitors' slender lead did not come under too much threat in the remainder of the opening period, and it took until the last kick of the half for the Reds to fire another half-chance at goal, but Joe Gomez's powerful drive flashed just wide of Leno's post.
Both coaches sent identical teams out for the second period, where Ryan Gravenberch took it upon himself to create the angle for a strike with a terrific forward surge in the 49th minute, but the Dutchman dragged his shot across goal and wide of the mark.
Liverpool's fast start to the second half soon faded, and a Fulham break in the 63rd minute should have seen the Cottagers double their lead, but Bobby Decordova-Reid inexplicably went for goal with Pereira poised for a tap-in and shot straight at Kelleher.
Reid would be left to rue that costly error just five minutes later as Jones had a crack from range, and the midfielder's effort took a wicked deflection off the back of Tosin Adarabioyo before looping into the back of the net past a stranded Leno.
After failing to penetrate the Fulham backline for over an hour, an attack-heavy Liverpool completed their turnaround within just three minutes of Jones's fortuitous effort, as Diogo Jota slipped in Darwin Nunez down the left, and the Uruguayan's cutback was met by Gakpo, who got in between Adarabioyo and Issa Diop to coolly finish with a first-time strike at Leno's near post.
The introductions of Gakpo and Nunez changed the game from a Liverpool perspective, and the former nearly turned provider for the latter in the 79th minute, but Leno was equal to his dipping header with an acrobatic save.
The German goalkeeper stuck a strong hand up to deny Nunez again in the 83rd minute as Liverpool sought to kill the game off, and the Liverpool number nine added another gilt-edged miss to his ever-growing collection in the 88th minute, shooting straight at Leno from inside the six-yard box after a pinpoint delivery from Bradley.
Prior to January 24's second leg at Craven Cottage, Liverpool are back in Premier League action away to Bournemouth on January 21, while Fulham only have two full days to recover before a lunchtime meeting with Chelsea in the West London Derby at Stamford Bridge.