The Hoops will not take part in the group stages of this season's competition, after losing 2-1 after extra time in the second leg against the Danish outfit.
The result saw them go out to a 3-2 defeat over two legs, which means they will play Czech side Jablonec in a Europa League qualifier instead.
"I take responsibility," Postecoglou said post-game per Sky Sports, referencing incoming transfers in particular.
"I'm the person who has been put in charge. We haven't got players in, I obviously haven't done a good enough job convincing people we needed to bring people in.
"I'm not going to shy away from it. I don't say that because I'm some kind of martyr, I just think that's my responsibility, that's why I was brought in.
"I have been trying to be as forceful as I can about what we need to bring in, and the challenges we have had are well chronicled.
"Irrespective of the result tonight, we still need reinforcements. We had a really young squad out there, young players on the bench. It's not a situation we need to be in."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!