Flower, whose contract is up at the end of the tour, and Cook received the backing of England and Wales Cricket Board (ECB) chief David Collier to retain their roles despite another miserable day which made a 5-0 series defeat highly likely.
"We need that experience," Collier said. "When you go through a transitional period you need somebody with knowledge, somebody who knows our system, somebody who works with all of our key coaches. Andy has all of those attributes and more."
He added: "Alastair fully deserves our support. People do grow into the job. Captains mature, players mature. We were convinced he was the right man at the time and we're still convinced today. We look forward to both Alastair and Andy leading us to success in the future."
When England were whitewashed in Australia seven years ago the ECB responded by conducting the Schofield Report, which looked at every element of the game. Collier ruled out the need for such a review this time around, but admitted there was a pressing need to "re-stock" for the year ahead.
"We are not going to do a review of that nature, but we will do a full debrief and learn the lessons that we need to learn from this tour," he said. "We have to look back and reflect on what has gone right and what has gone wrong and learn lessons from it."
After their top order failed once again, England were bowled out for 155 yesterday, before Chris Rogers (73no) helped Australia to 140 for four and a lead of 311 runs at the close of day two in the fifth and final Test. That followed humiliating defeats in the first four Tests of the series and Gooch admitted the tourists had simply not played well enough to compete, admitting all personnel, including himself, must prove they are capable of much better if they are to earn the right to stay on after the "debris" of their Ashes tour.
"Everyone has to look at themselves," he said. "That's the coaches, the players, they're all going to be under scrutiny, quite rightly, and we all have to take it on the chin and we have to take the criticism. If you play the way we've played, the brutal truth is it's not good enough.
"We have to look at ways to move forward and we have to look to ways to improve even though that might entail taking some more pain before it gets better.
"There will be a rebuilding process, I think that's fair to say."
On an awkward pitch, Cook's lbw dismissal to the second delivery of the morning to Ryan Harris set the tone. Harris, Mitchell Johnson and Peter Siddle each took three wickets, and only Ben Stokes provided any worthwhile resistance in an England innings which yet again fell well short of requirements.
They lost four wickets for 15 to stumble to 23 for five - and hard as Stokes battled, in stands of 39 with debutant Gary Ballance and 49 with Jonny Bairstow, it was merely a continuation of similarly shambolic performances in each of the preceding Tests.
"Our players have not met the challenge with the bat, obviously," Gooch said. "We've not competed as we'd have liked to and not shown the skills that have been necessary ... we've gifted too many wickets. We have only one hundred in four-and-a-bit Tests, and that's not going to win you anything."
Cook's shot, or lack of it, was that of an individual bearing the weight of responsibility in adversity.
"When things have gone the way they have ... there is more pressure, and you feel more responsible as a captain," said Gooch. "I know that feeling. I wouldn't say he made that mistake purely because he's captain and he's under pressure, but it's been a problem for us all through the tour. You make mistakes and sadly we've made too many."
Cook is one of several senior batsmen who have been unable to raise their game, as Australia have taken complete control. Gooch cited very good Test match bowling from the hosts as one reason, but said as well: "You'd still expect players of that calibre, with that record behind them, to score a proportion of runs - and they've not been able to do it. If things don't improve then everyone's position is going to be under scrutiny. That goes for players. If you don't perform, ultimately you get left out. Everyone [here] would not like to leave under these circumstances ... and the debris of this tour. I can't speak for everyone, but I'd be surprised if everyone didn't want to put things right."