Woodward, who coached England to their first ever victory on Australian soil in 2003 and followed up with their second in the World Cup final that year, had said he felt the Wallabies could wilt under pressure in the Pool A game at Twickenham.
"If you can keep hold of the ball and run through the phases, always moving forwards, Australia will disintegrate," Woodward wrote in Friday's Daily Mail.
"Contrary to popular belief, they are not the brightest team, they give away penalties and pick up yellow cards when they are stretched."
Asked about the comments, a smiling Cheika told a news conference: "Mr Woodward is right. I only got 300 out of 500 in my high school certificate. My mother wasn't happy either with the result, I can assure you, but somehow I got through.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"But no, sometimes it's not intelligence but emotional intelligence that helps in this game. It's not the IQ, it's the EQ."
Cheika, fluent in English, Arabic, French and Italian and who ran a multi-million-dollar fashion business before setting out on a hugely successful coaching career, took it all in good spirits.
"I understand it's all a bit of fun and games and a bit of a snipe here, a snipe there, I'm pretty comfortable with all that stuff," he said.
"I believe a lot in my own players so I don't think having a crack at other players, it's all a bit of show, I don't think that's what I am here to be doing, others can do that.
"But for us it's almost letting down our country if we don't perform at least with the intent to be at our maximum intensity every time we play.
"For someone who never played for Australia how much that means and how every moment in that jersey should be cherished because you're blessed to be playing and so fortunate and every game should be like it's your last in the jersey."
Winger Adam Ashley-Cooper, a survivor from the teams' last World Cup meeting when England triumphed in the 2007 quarter-finals, was equally upbeat.
"I think with the build-up and it being a highly anticipated test match it's very exciting for all of us to be a part of it," he said.
"We've had a great rivalry over the years that I've certainly been a part of. Test matches against England have always been of high intensity and high quality so I'm sure its going to be no different."
(Editing by Clare Fallon)
