Gazidis addressed fans groups at a meeting on Wednesday night - which had been arranged before the shock 3-2 penalty shoot-out exit 24 hours earlier at Valley Parade.
"I think I am frankly tired of getting up here and delivering the same message," said Gazidis. "Tuesday night was not good enough and it made us all upset and angry.
"I would like to apologise to all of you, especially the fans who travelled up there. It was unbelievable support as ever and you deserved better.
"That is something we will work hard to put right. We all work here and are desperate to deliver the success and trophies we all want."
Meanwhile, the Arsenal Supporters' Trust claim under-fire manager Arsene Wenger will be given money to spend in the January transfer window.
"There is plenty of money there," said an AST spokesman. "That was the message that Ivan and his finance director were giving this evening."
But Gazidis maintains Arsenal are not about to just buy their way to success - and claims the fans would not have it any different.
In a wide-ranging interview with the January edition of Four Four Two magazine, the Gunners chief executive insisted the club's self-sustaining business model will pay off in the long term.
"If you do any analysis of performance against spending over the past 15 years, Arsenal is the one club which has outperformed its spend every single year over that period," Gazidis said.
"That does not happen by accident; it happens because we have an outstanding manager who is an outstanding judge of talent and very experienced in how to manage a team.
"There are some real fundamentals at this club which I think our fans believe in - we stand on our own two feet and take pride in our achievements.
"We are not reliant on anyone else for success. That is the model our fans believe in, and for good reason."
Last month, Arsenal announced an extension to the sponsorship deal of Emirates Airline, which will be worth around £30million a year.
Gazidis, however, insists simply throwing all of that cash at the team is not the answer for a quick fix.
"I am not sure what the alternative strategy is, it has not really ever been spelt out," he said.
"I assume the alternative strategy is to spend more than the nation state of Abu Dhabi, more than anyone else, and I am not even sure I would describe that as a strategy, but it is the only other thought that comes to mind."
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