Gazidis met with fans' groups on Wednesday night following the quarter-final exit at Valley Parade on Tuesday which has heaped further pressure on manager Arsene Wenger.
"Tuesday night was simply not good enough and it made us all upset and angry," he said in quotes reported by several national newspapers.
"I would like to apologise to all of you (the supporters), 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. Look into the eyes of the staff here, look at the way they all talk. We will get this right."
Arsenal Supporters' Trust spokesman Tim Payton also said Gazidis had promised "significant funds" were available to strengthen the squad. Payton added:
"This is the most difficult time in Arsene's 16 years and both the club and himself need to change direction quickly if his wonderful era is not to end on a sad note.
"The Arsenal Supporters' Trust cannot fathom why £70million was left untouched this summer. Arsene needs to get over his distaste for spending and use the resources that after all are provided by the hard-pressed fans who watch the team.
"The AST believe a shake-up is needed. We would like to see new, younger blood added to the board, and they also need to exercise more control and direction over Arsene's football strategy." The Frenchman has been described by former Arsenal midfielder Stewart Robson as "a dictator in many ways". Robson claimed that Wenger's backroom team were unable to challenge his decisions. As for the performance against Bradford, one of the Bantams players, Gary Jones, compared Arsenal unfavourably with npower League Two rivals Torquay who they played last weekend.
"I think Torquay gave us a tougher game," Jones was quoted as saying in several national newspapers this morning.
"It was like a role reversal against them. Torquay defended well against us and we defended really well against Arsenal. The lads were unbelievable."
Asked if Arsenal should have been embarrassed by the result, Jones added:
"Without a shadow of a doubt. No disrespect to us and our lads but they should be beating Bradford City."