After a Micky van de Ven first-half own goal gave Fulham a deserved lead, Spurs levelled in unusual circumstances.
The home side were temporarily reduced to 10 men when Kenny Tete had to change his boot and Spurs took full advantage when Richarlison headed in.
But Davinson Sanchez's penalty was saved as Fulham won the shootout.
It was Tete - with a new boot on - who scored the decisive spot-kick to send the home side through.
It was ultimately a deserved win for Fulham, who should perhaps have been out of sight before Richarlison levelled with his first goal of the season after the hosts dominated the first half against a sluggish Spurs side.
Tottenham improved after the break and Manor Solomon almost got a dream winner against his former side late on, but his curled effort from the edge of the box was saved.
Substitute James Maddison also hit a free-kick into the wall but penalties were needed to settle the tie, with Fulham successfully converting all of theirs to send Spurs out.
It was overall a very encouraging performance by Fulham, whose manager Marco Silva had to watch the game from the stands after picking up three yellow cards in the first three Premier League games of the season.
Richarlison relief as he gets off the mark
Consequently, this EFL Cup match offered Spurs boss Ange Postecoglou the chance to rotate his squad and give others the opportunity to impress.
Richarlison was one of just two players - along with Van de Ven - to retain his place in the side from the 2-0 win against Bournemouth, and this was a chance for the Brazilian to finally get off the mark for the season as he looks to fill the void left by Harry Kane's departure.
It looked like another opportunity would pass him by as he was largely anonymous in the first half, having one shot of note that was easily blocked by the Fulham defence.
But he showed his predatory instincts to nod home a fine cross from Ivan Perisic and, despite this defeat, Postecoglou will hope this will give him the confidence to kick on and start scoring in the Premier League.