When Bobby Decordova- Reid headed the away side ahead on 80 minutes, Jurgen Klopp’s team appeared doomed, but goals by Wataru Endo on 87 minutes and Trent Alexander-Arnold a minute later turned the game around.
Alexander-Arnold’s free shot hit the crossbar and rebounded in off Former Red Harry Wilson scored for Fulham after Bernd Leno put the hosts ahead.
Alexis MacAllister’s spectacular long-range shot restored the Reds’ lead, but Fulham equalized in first-half stoppage time when Kenny Tete scored.
Decordova-Reid appeared to have won it for Marco Silva’s men, but substitute Endo and vice-captain Alexander-Arnold had different plans.
The triumph extended Liverpool’s unbeaten record in the Premier League to seven games, leaving them two points behind league leaders Arsenal.
Alexander-Arnold was the hero at the end, and it appeared he’d started the ball rolling for Liverpool when his free-kick found the back of the net on 20 minutes.
The Premier League later stripped him of the goal as the ball hit the bar and struck Leno before dropping into the net.
But any resentment he felt at having his goal disallowed was washed away in the 88th minute when his powerful drive from the edge of the box soared into the net to give the 2019/20 champions what may be a massive three points.