It is their earliest League Cup exit since 2005. For both clubs, this was undeniably a tough draw but Spurs are now out of the season's most winnable competition before the top seven have entered the draw and, with August still not over, their chances of ending a 15-year wait for a trophy already hinge on the FA Cup, barring a Premier League campaign for the ages.
Postecoglou has been a breath of fresh air since taking the job and made a hugely encouraging start in the League, but the Australian has questions to answer over his selection on the night.
Despite the lack of European football this season, the head coach made nine changes from Saturday's win over Bournemouth, with only Richarlison - who opened his account for the season with a second-half equaliser - and Micky van de Ven keeping their places.
The result was a disjointed performance - 'Ange Ball' in name only - and one which suggests some of these players may not have what it takes to play the head coach's high-octane passing game long-term.
Given Spurs' schedule, making so many changes for the game was strange, threatening the positive mood and momentum which has built up around the club following seven points from their first three League fixtures.
Postecoglou will reason that his fringe players need minutes and a chance to play their way into his plans (or into a big move before the transfer deadline), but making so many changes in one burst rarely makes for a coherent display, especially when Fulham went with more of a full-strength side, albeit with six changes from their 2-2 draw with Arsenal at the weekend.
Going forward, Postecoglou will have even fewer chances to give minutes to his fringe players, with Spurs facing a maximum of 45 games this season if they reach the FA Cup final.
Van de Ven was Spurs' standout player, at least until the late introduction of James Maddison, and was desperately unlucky to score a second own goal in four matches, deflecting Tom Cairney's cross past Fraser Forster after the Fulham captain had turned Davinson Sanchez inside out.
Sanchez particularly struggled with and without the ball, while the midfield three of Giovani Lo Celso - making his first Spurs start since 2021 - Pierre-Emile Hojbjerg and Oliver Skipp struggled to play through Fulham's excellent press.
The Colombian was also the villain of the shootout, missing the only spot-kick of the eight when his low effort was comfortably saved by Marek Rodak.
It was a strange call to have him take third when Maddison (who went fourth) and Hojbjerg (not selected) were among the players still to come.
Postecoglou replaced Lo Celso with teenager Dane Scarlett at the interval - perhaps speaking to his desire for a new senior striker in the next few days - and Spurs did improve, Richarlison heading home Ivan Perisic's cross in what could prove a crucial goal for the struggling Brazilian.
A raft of later changes, including Maddison and Dejan Kulusevski, made a positive difference but it was Fulham who continued to look the more threatening, with Bobby De Cordova-Reid missing the best of their second-half chances.
For Spurs, an early cup exit is a gut punch, leaving them with nothing but League fixtures until January.