Davinson Sanchez was the only player to miss in the shootout, which was wrapped up by a cool finish by Kenny Tete.
Micky van de Ven turned into his own net from a Tom Cairney cross to give Fulham a deserved lead in the first half but they were pegged back by Richarlison's back-post header just before the hour.
Fulham shaded the chances but neither side could find a breakthrough in the 90 minutes.
Dan Kilpatrick was at Craven Cottage to rate the Spurs players...
Fraser Forster 6
Made a couple of important - if routine - saves to keep Spurs in the game. Did not get close to any of Fulham's penalties and not fully comfortable playing out from the back.
Emerson Royal 6
Defended his back post well but offered little in the way of attacking impetus.
Davinson Sanchez 4
Struggled with and without the ball in a desperate first-half display, including being turned inside out by Cairney for the Fulham goal. But improved after the break and denied Harry Wilson with a crucial block. The unfortunate villain in the shootout.
Micky van de Ven 7
Stood out and was unlucky to turn the ball into his own net for the second time in four matches. Appeared to relish the very different physical battles with Rodrigo Muniz and Adama Traore.
Ben Davies 5
The captain for the night made no impact in possession but was solid enough defensively.
Pierre-Emile Hojbjerg 4
Really wasteful in possession and unable to play through the Fulham press. Missed a glorious chance to level the game when the ball fell to him in the box.
Oliver Skipp 5
Ran tirelessly but loose with the ball in the face of Fulham's well-organised press.
Giovani Lo Celso 6
Scampered around busily but struggled to make an impression in the final third and was often forced deep to pick up the ball.
Manor Solomon 6
Showed quick feet on the ball and forced Marek Rodak into a late save with a trademark curling effort on his return to the Cottage.
Ivan Perisic 7
Made Richarlison's goal with a superb cross to the back post.
Took his goal well, having struggled to get into the game for the first hour. Can it kick-start his season - and Spurs career?
Dane Scarlett (Lo Celso 45') 6
Helped to stretch Fulham's back line at the start of the second half but never looked a goal threat.
Heung-min Son (Perisic 71') 5
Hesitated when put through 1v1 with a defender in the box and the chance was gone.
Dejan Kulusevski (Richarlison 71') 6
Released Son for a late opening and created a half-chance for Scarlett with a smart run.
Pape Matar Sarr (Skipp 71') 6
Added fresh legs to the midfield battle, where Fulham were largely superior.
James Maddison (Solomon 82') 6
Immediately made a difference with a weaving run to the byline but his cross was behind Kulusevski.
Not used: Vicario, Romero, Udogie, Bissouma.