Milne congratulated the Fifers and boss John Robertson for their win.
And he said: "It is my job, and that of my fellow board members, to support the management team in whatever way we can to turn things round."
The advent of Aberdeen exiting cup competitions to lower league opposition has become commonplace in recent years but Tuesday's defeat was a first such defeat for Brown and assistant Archie Knox at Pittodrie.
“Everyone at the club needs to take a long hard look at themselves, at the preparation ahead of games, what we could do differently and how we can improve”
Stewart MilneAberdeen chairman
Brown, whose side are 10th in the Scottish Premier League, admitted the loss was the "worst result of my managerial career" but vowed to fight on.
"I have got confidence in my own ability to turn things around," said Brown. "I think Archie and I are capable of that.
"We didn't play at all. I possibly made a mistake by changing the team too much and leaving out good players from the start thinking we had enough in reserve to cope with this fixture.
"I have never considered my own position at all and even after [the defeat by East Fife] I'm determined.
"If other people consider it there is nothing I can do about that - I just have to get on with my job conscientiously."
Milne added on Aberdeen's official website: "The manner of the defeat was totally unacceptable and I would like to take this opportunity to say sorry to all of our supporters who most certainly do not deserve to have to endure what was yet a further thoroughly humiliating experience.
"Quite simply, we should have had more than adequate resources to win the game.
"I truly believe the squad we have at our disposal is comparable to any in the SPL outwith the Old Firm but, for whatever reason, too many of them are performing at levels way below what they are capable of.
"When we were looking for a new management team less than a year ago, Craig and Archie were top of our list of candidates and we went out of our way to ensure we brought them to Pittodrie as we firmly believed they were the duo to take this club forward.
"Nothing that has occurred subsequently has changed my opinion, although last night's result was a massive setback.
"Everyone at the club, including myself, the manager and the players, needs to take a long hard look at themselves, at the preparation ahead of games, what we could do differently and how we can improve to ensure there is no repeat of the type of feelings we all experienced last night."
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