Postecoglou defends selection after Spurs crash out of League Cup
Postecoglou had masterminded an encouraging start to his first season in charge of Tottenham, with successive Premier League wins against Manchester United and Bournemouth following a draw against Brentford.
But the former Celtic boss paid the price for making nine changes to his starting line-up at Craven Cottage.
Postecoglou handed a full debut to Manor Solomon, while Giovani Lo Celso and Pierre-Emile Hojbjerg started despite speculation over their futures.
Fulham boss Marco Silva made six changes himself, but it was Tottenham's understudies who struggled.
Micky van de Ven's 19th-minute own goal gave Fulham the lead as the defender turned Tom Cairney's cross past keeper Fraser Forster.
Tottenham took advantage of a curious incident that left Fulham down to 10 men to equalise in the 56th minute.
A problem with Kenny Tete's boot meant the Fulham defender had to sprint down the tunnel to find a replacement in the dressing room.
Moments later, Ivan Perisic whipped in a cross from the position Tete would have been defending and Richarlison headed home from an acute angle.
But Tottenham were unable to find a winner and in the shoot-out Davinson Sanchez's effort was saved by Marek Rodak before Tete, now with boots on both feet, fired the decisive kick.
Currently enduring a trophy drought that stretches back to 2008, Tottenham already have one less competition in which to end that barren run this term.
Postecoglou had introduced stars James Maddison and Son Heung-min in the second half, but defended his decision to select his fringe players.
"How am I going to find out about my players? It doesn't mean the cup, or any game, isn't a priority for us," the Australian said.
"We are very much at the discovery stage and we need to give the players the opportunity.
"We are going to need these guys if we get disruptions along the way with injuries."
'Absolutely incredible'
Fourth-tier Salford, co-owned by former Manchester United stars David Beckham, Gary Neville, Ryan Giggs and Paul Scholes, beat Leeds 9-8 on penalties after a 1-1 draw with the Championship club.
"Absolutely incredible. I would have never imagined this game to go so far, never mind to go to penalties and to win. Such a big night for all these Salford fans," Neville said.
Hoping for a reunion with his old club in the third round, Neville added: "Us lads as owners, we love Man United and we love this club as well.
"To have the opportunity to play them is the special night that we want."
Luton beat fourth-tier Gillingham 3-2 at Kenilworth Road to win for the first time this season.
After losing to Brighton and Chelsea in their first two Premier League games, Jacob Brown, Alfie Doughty and Cauley Woodrow scored for Luton, who are playing in the top-flight for the first time in 31 years.
Bournemouth boss Andoni Iraola earned his first win in English football as Ryan Christie's last-gasp goal clinched a 3-2 victory at second-tier Swansea.
The Spaniard had overseen a draw with West Ham and defeats by Tottenham and Liverpool in his first three league games.
Brentford needed a 3-0 penalty shoot-out victory to advance after a 1-1 draw at fourth-tier Newport.
Crystal Palace survived a scare as they fought back from two goals down to win 4-2 at second-tier Plymouth thanks to Jean-Philippe Mateta's hat-trick.