A 19th-minute own goal from Micky van de Ven put the hosts deservedly in front against a lacklustre Spurs side, who managed to restore parity 11 minutes after the break courtesy of a close-range header from Richarlison.
However, it was Fulham who has the last laugh as they converted all five of their efforts in the penalty shootout, while Davinson Sanchez was the unfortunate one who saw his tame strike from 12 yards saved by Marek Rodak.
While Fulham made amends for their shock elimination to Crawley Town in the second round last season, Tottenham have crashed out as early as Round Two for the first time in 18 years.
Ange Postecoglou made nine changes to the Tottenham side that beat Bournemouth 2-0 in the Premier League last weekend, with only Van de Ven and Richarlison retaining their places, while former Fulham man Manor Solomon was handed his full debut.
Without the presence of Marco Silva on the touchline due to a one-game suspension, Fulham made six changes of their own after drawing 2-2 with Arsenal last Saturday, with the two most notable inclusions being new recruit Adama Traore - who was handed his first start - and striker Rodrigo Muniz.
It was the hosts who began on the front foot and the first chance fell to Muniz at the back post in the seventh minute, but he was unable to head Antonee Robinson's looping cross from the left flank on target.
Fulham's persistence eventually paid off, though, and they opened the scoring on the 19-minute mark when Van de Ven turned a low cross from Tom Cairney into his own net, with Cristian Romero unable to clear the ball off the line at the far post.
While the Tottenham away end vented their frustration at chairman Daniel Levy, the Spurs backline were struggling to deal with the pace of Traore, who teed up Muniz following a swift counter attack just after the hour mark, but the Fulham forward's low-driven strike was beaten away by Fraser Forster.
Forster denied Muniz again on the stroke of half time, diving to his right to palm away a powerful header directed towards the bottom corner.
Postecoglou opted against bringing on the big guns such as Son Heung-min and James Maddison at half time, and instead called upon 19-year-old striker Dane Scarlett, who replaced Giovani Lo Celso, while Fulham unsurprisingly were unchanged after the interval.
There was little change to the dynamics of the contest in the early exchanges of the second half, with the Spurs fans growing increasingly frustrated with their side's performance.
However, the visitors sparked into life on the 55-minute mark when Pierre-Emile Hojbjerg fired an effort narrowly wide, just a few minutes before Richarlison headed home an equaliser, rising at the back post to nod a cross from Ivan Perisic across the face of goal - his first goal for Tottenham since September last year.
Spurs' equaliser came just seconds after Fulham's right-back Kenny Tete bizarrely ran down the tunnel to change his boots, briefly leaving the hosts down to 10 men as the visitors capitalised on the numerical advantage.
Fulham responded well and came close to retaking the lead when Bobby De Cordova-Reid sliced an effort inside the penalty area narrowly wide, shortly before Postecoglou called upon Son, Dejan Kulusevski and Pape Matar Sarr for the final 20 minutes, with Richarlison, Perisic and Oliver Skipp making way.
Maddison and Raul Jimenez came on for either side in the closing stages as the game gradually became stretched, and although both teams pushed for the winner, the tie was eventually forced to penalties.
Fulham trio Andreas Pereira, Jimenez and Harry Wilson all comfortably dispatched their efforts from 12 yards amidst strikes from Spurs duo Son and Kulusevski, before Rodak stretched low to his left to keep out a powerless strike from Sanchez to put the hosts in control of the shootout.
Joao Palhinha and Maddison then both made the net ripple to mount pressure on Tete for the decicive spot-kick, but he managed to slot home his effort with aplomb to send Spurs packing and Fulham through to the third round.
Fulham will learn their third-round opponents when the draw is made on Wednesday evening.