Postecoglou made nine changes from the 2-0 wins over Bournemouth and Manchester United, with only Richarlison - who scored Spurs's equaliser in a 1-1 draw after normal time - and Micky van de Ven keeping their places.
Fulham deservedly took the lead in the first half when Van de Ven turned Tom Cairney's cross into his own net and the hosts had the better of the chances before spot-kicks.
Davinson Sanchez missed in the shootout, Marek Rodak saving the Colombian's low effort to confirm Spurs' earliest League Cup exit since 2005.
"[The League Cup] is my priority but there is no European football, so how am I going to find out about our players?" Postecoglou said afterwards.
"What opportunity would I have to do that other than the game? They are all part of our club. We thought they'd be able to bring a different energy tonight, and that doesn't mean the cup or any game isn't our priority.
"From my perspective, we're very much at the discovery stage so we need to find out and give the players the opportunity to contribute. Because that's why they're here - they're not just here to make up the numbers. They keep working hard and deserved the opportunity to go out and play tonight.
"Obviously I'm disappointed with the outcome. We weren't at our fluent best particularly early on which was not surprising considering the amount of changes I made. A lot of the boys tonight, it was their first start of the season.
"We lacked cohesion and fluency. But I really liked the way we hung in there. There are many different ways to grow as a team and we showed some real character tonight. To hang in there and we had our moments in the second half to maybe go and win it. We weren't able to get over the line and lost on penalties."
Richarlison headed home Ivan Perisic's cross shortly before the hour to open his account for the season and potentially kick-start his stuttering Spurs career.
Postecoglou says he is confident more goals will come for the £60million striker and is more focussed on improving his all-round game.
"I guess because everyone's been talking about it it's great for him to score a goal but I think he's played better for us in the other games," Postecoglou said.
"I get why other people look at that and I understand that but for me it's about contributing to our football, how we play. I thought we gave away the ball a bit too much tonight, not just him but a few players in that boat.
"It's an important part of our game and that's the stuff I want Richy to improve on. I think the goals will come. Our strikers, our attacking players will get opportunities to score. Our all-round games is much more important at this stage."
The Australian introduced teenager Dane Scarlett for Giovani Lo Celso at half-time and said he is still to decide if the 19-year-old forward will be going out on loan before Friday's 11pm transfer deadline.
"He played a fair few games last year and I think he's improved as a player since then," Postecoglou said. "I'll make that decision over the next couple of days and sit down with him, but I certainly see him as a Tottenham player and a lot of that will depend on what else happens over the next two or three days."