One week on from their astonishing 4-0 battering of reigning champions Manchester City, the Lilywhites were sent crashing back down to earth as they played out their first home stalemate in the top flight since April 2023.
Brennan Johnson put the Lilywhites ahead - arguably against the run of play - early in the second half, but Fulham skipper Tom Cairney then came off the bench to strike a terrific leveller for the visitors.
Cairney was then dismissed in the closing stages for an awful tackle on the back of Dejan Kulusevski's leg, forcing Fulham to see out the final 15 minutes with 10 men, but Postecoglou's men could not force their way through even with a numerical advantage.
The Lilywhites' offensive efforts were not helped by a fresh selection blow before the game, as Dominic Solanke rocked up with an illness and was sent home, forcing Postecoglou to put out a front three of Johnson, Timo Werner and makeshift striker Son Heung-min.
Will Solanke return for Bournemouth game?
Solanke was one of four Spurs attackers to miss Sunday's derby, alongside fellow illness victim Mikey Moore, Richarlison (hamstring) and Wilson Odobert (hamstring), although Postecoglou is optimistic that he will be able to take on former club Bournemouth in midweek.
"I hope so," Postecoglou replied when asked if Solanke could be back on Thursday in his post-game press conference. "He trained yesterday but he came in today and didn't look well at all so we sent him home. It's obviously an illness not an injury so we need to see how it settles down in the next couple of days but hopefully it's nothing too significant."
While one of Fulham's subs in Cairney came off prematurely due to his red card, one of Tottenham's second-half changes - Archie Gray - was withdrawn in the closing stages after succumbing to an injury.
The teenager did not look at all comfortable at the end, but Postecoglou does not think that he will be missing for long, saying: "He got a knock, a bit of a dead leg so hopefully nothing too significant."
Gray had initially come on to help shore up a Tottenham backline still missing Cristian Romero, Micky van de Ven and Guglielmo Vicario, the former of whom did not recover from a toe problem in time to make the derby.
Romero yet to return to team training with Tottenham
As a result, Ben Davies, Radu Dragusin and Fraser Forster - who put in a terrific display in between the sticks - formed the spine of Spurs' defence, but Postecoglou has not ruled Romero out of the Bournemouth showdown on December 5.
Asked whether Romero might make the cut on Thursday, Postecoglou added: "Potentially mate. He still hasn't trained with the team, so he's getting closer but he still hasn't trained with the team. So, him and Micky are still working with sport science staff. So, hopefully, not too far away."
On Forster's display, Postecoglou talked up the importance of having an experienced goalkeeper not overwhelmed by stepping in for the resident number one, saying: "Yeah, he was good.
"We needed him obviously in some big moments today, and I think when you lose your goalkeeper and somebody as significant as Vic for us, I think the key to that is to have somebody experienced who's not going to be sort of overawed by filling those shoes.
"Fraser, certainly, he's done it for a number of years. We see him every day. What you saw out there, we see every day training and he still has a really strong work ethic and we had a couple of big moments today which were important for us and again, he will be important moving forward."
Spurs' first home draw in 27 Premier League games leaves them seventh in the top-flight standings on 20 points, three below Brighton & Hove Albion in the final Champions League spot.