Bobby Decordova-Reid had seemingly done enough to end the Reds' perfect Anfield run, but late strikes from Wataru Endo and Trent Alexander-Arnold completed a pulsating Liverpool fightback, forcing Marco Silva's dumbstruck side to travel home empty-handed.
Few surprises were sprung by Klopp, who reverted to his strongest XI after their Europa League beating of LASK Linz, while Fulham boss Silva curiously demoted Willian to the bench in favour of ex-Liverpool man Harry Wilson, despite the Brazilian's brace against Wolverhampton Wanderers.
It took just three minutes for Leno to get his gloves dirty, as the Fulham goalkeeper did well to hold onto a Luis Diaz drive, five minutes before the offside flag denied Mohamed Salah his landmark 150th Premier League goal.
Diaz - who committed the offside offence - had left Leno in a heap on the ground in the build-up to Salah's disallowed strike, but the ex-Arsenal man was cleared to continue after the medical team had encased his head in a black bandage.
Leno also donned a swollen right eye as he soldiered on, but there was nothing that any clear-headed goalkeeper could have done about Liverpool's deserved opener in the 20th minute, as Alexander-Arnold's venomous free kick rattled the bar and bounced into the roof of the net via Leno's back.
Joao Palhinha was the guilty party, tripping Dominik Szoboszlai in Alexander-Arnold range, but just four minutes after the right-back's wonderful strike and Leno's unfortunate own goal, Silva's eyebrow-raising decision to introduce Wilson over Willian was vindicated.
The Cottagers were allowed to build up down the left through Alex Iwobi and Antonee Robinson, and the latter's low cross into the six-yard box found a charging Wilson, who evaded Mac Allister all too easily and poked home through the legs of Caoimhin Kelleher at the near post.
Fulham suddenly had their tails up at Anfield, only to be undone by another outrageous attempt in the 39th minute, as Mac Allister - making up for his role in Wilson's equaliser - latched onto Raul Jimenez's misplaced header 30 yards from goal and unleashed an extraordinary thunderbolt into the top corner.
The Argentine's strike was certainly worthy of a goal-of-the-season nomination alongside Alejandro Garnacho and Michael Olise's stunners from last weekend, but owing to the earlier injury to Leno, Fulham were gifted nine minutes of additional time to claw themselves level again.
The visitors did exactly that with six minutes of injury time gone, as Jimenez nodded a corner into the corridor of uncertainty, where Tete - back in the starting lineup - managed to squeeze the ball past Kelleher; the offside flag went up at first, but a VAR review awarded Fulham their second equaliser of the afternoon.
Silva's side had the ball in the net for a third time towards the end of the half, as Tim Ream capitalised on more lackadaisical Reds defending and pounced on a saved Palhinha header, but the Cottagers centre-back was offside this time.
The second half began in a similarly frenetic vein, with Darwin Nunez rattling the crossbar in the 53rd minute before the Uruguayan added another horror miss to his collection, firing wide of Leno's goal after being played through by Salah's header on the hour mark.
At the other end of the field, Klopp was forced into a defensive alteration as Joel Matip came off injured to be replaced by Ibrahima Konate, and the Reds coach was left to bemoan more sloppy play from his side as Fulham took a shock lead with 80 minutes played.
An overhit Decordova-Reid cross from the right was salvaged by Willian, who slipped through Tom Cairney in the left-hand side of the box, and the Cottagers captain chipped a delicate ball to the back stick, where Decordova-Reid towered above Kostas Tsimikas to head home.
However, what Liverpool lacked in defensive nous, they made up for with scintillating attacking play, and Endo brought the Reds level just seven minutes later, picking out the top corner with panache from inside the D after strong hold-up play by Salah.
Just two minutes after the restart, Anfield erupted into unbridled chaos as Liverpool restored their one-goal lead through Alexander-Arnold, who may have been denied the recognition for his astonishing free kick, but there was no preventing the vice-captain from bagging his rightful goal.
Leno initially did well to deny a low Cody Gakpo strike, but Nunez kept the ball alive and sent a delivery to the back post, which was only headed away as far as Alexander-Arnold, and the right-back crashed home a vicious half-volley from just inside the box.
Setting a new club record for their latest-ever Premier League comeback win, Liverpool have temporarily risen above Manchester City into second in the table before the champions take on Tottenham Hotspur, while the Cottagers stay 14th, eight points clear of the relegation zone.
Liverpool now have a date with Sheffield United at Bramall Lane pencilled into the diary for Wednesday evening, while Fulham host Nottingham Forest at the same time in midweek.No Data Analysis info