England Rugby: We're just not good enough
After being edged out by Wales 28-25 last weekend, England knew nothing but a victory would do at Twickenham against Australia, but the Wallabies exposed nerves and frailties at the breakdown to batter the hosts 33-13.
"Let's not waste time with excuses. Let's lose with dignity," Woodward said in the Daily Mail.
"In the final analysis we simply haven't been good enough, on or off the field, at this World Cup and the way forward will be just that little bit easier if we admit that straight away."
Woodward, who masterminded England's triumph in 2003, added: "It was a tough pool but England should have been well capable of beating both Wales and Australia, familiar opponents they have beaten many times in recent years.
"No expense has been spared in England's preparation and they were at home in both matches. Everything was in England's favour and they should have cashed in."
Waratahs fly-half Bernard Foley had an evening to remember as he scored 28 points - with two tries - to give Australia their biggest ever win at the home of English rugby.
Matt Giteau delivered the killer blow with a try late on after Anthony Watson had given England, who are now the first host nation to be knocked out before the knockout phase, a glimmer of hope.
Lancaster admitted he will consider his future after watching four years of hard work go up in smoke, but Woodward wants the recriminations to wait until after the tournament is over.
"The future of Stuart Lancaster and others must not be allowed to hijack this tournament," he said.
"There will be an inquest and consequences for those in charge - and those who made the appointments - but for the remainder of the competition itself I would like to see all concerned go to ground and keep their counsel."
Martin Corry, a member of Woodward's successful 2003 squad, also wants the dust to settle before Lancaster's future is decided.
"There is a state this morning of numbness, of disbelief," the former lock told BBC Radio 5 Live.
"Let's not have one of these quick, knee-jerk reactions. Let's take the time and have a look and then when emotion goes, let's make the decisions.
"It's wrong to go 'right, we're out of the World Cup, everything has to change'."
Another former World Cup-winning coach, New Zealand's Graham Henry, believes something of an identity crisis contributed to England's failure.
"I think they've been playing the wrong game for a long time," he was quoted as saying by the New Zealand Herald.
"They are trying to simulate the way the All Blacks play and I don't think they have the skill level to do that because they simply don't come from that sort of environment.
"They've changed their game in recent years and it hasn't been successful so Lancaster will get some stick in the press. He's tried hard Stuart, no doubt about that. He has good standards in his team, good culture but they're not good enough at this level."