England came into the day 202 runs behind Australia, but enters the final day leading by 88 with Cook unbeaten on 132 and Jonathan Trott on 54.
Strauss and Cook shared a 188-run opening stand to produce England's best ever partnership at the Gabba ground before the England captain was dismissed for 110, stumped by Brad Haddin off Marcus North.
Cook needs seven more runs to record England's best score at the Gabba and surpass Ian Botham's innings in 1986-87.
The opening partnership surpassed the stand of 160 by Graeme Hick and Graham Thorpe in 1994, and more notably Jack Hobbs' and Herbert Sutcliffe's mark of 3,249 to become England's most prolific opening partnership.
"Those records come because the two of you are able to stay in the side," Strauss said. "You need a certain resilience to do that. It's one of Cookie's greatest strengths."
Strauss was dominant in the morning session, being particularly severe on Shane Watson who he hit for four boundaries in five overs. He brought up his first century in Australia with a superb late cut off Xavier Doherty.
Strauss said the failure of his first innings, when he was caught for a duck off his third ball, drove him to succeed today.
"It was pretty much the worst I've felt on a cricket pitch," Strauss said. "But that is the wonderful game of cricket. It reminds you you have to respect the game."
Strauss also admitted to a bout of nerves after the first ball of the second innings, when he narrowly avoided an lbw decision to Ben Hilfenhaus.
"I thought it was a very good leave," Strauss said. "My heart was definitely in my mouth. I thought it was a bit high and thankfully that was the case."
Strauss seemed to lose his rhythm after reaching his century, scoring just 10 from his next 40 deliveries before a rash cross-batted shot caused his demise.
Cook looked less assured, but ground his way to his second century against Australia.
"The conditions at home during the summer were the toughest I've ever faced," Cook said. "It reminds you that when conditions are better and it isn't swinging, it's time to cash in.
"That noise when I got a hundred was a great moment. It made the hairs on the back of my neck stand up."
Although England toiled in the field for much of the previous day against centurions Mike Hussey and Brad Haddin, their bowlers produced a much better comeback at the end of the day, taking five wickets for 31 runs. In contrast, Australia's bowling attack failed to make an impact today and, despite overcast conditions, there was no discernible swing.
Strauss said his side would still treat the Australian bowlers with caution. "We respect all their bowlers and we'll continue to do that," he said.
In a day of few opportunities for the Australians, Trott gave a slight chance to Michael Clarke at point off the bowling of Peter Siddle.
Australia's hero in England's first innings, Siddle was also unlucky in the morning session as Cook edged just wide of third slip and later Strauss missed his leg stump from an inside edge.
Strauss was given a reprieve before lunch while on 69 when he hit a Doherty ball to Mitchell Johnson at mid-on, but the chance was dropped.
England scored 260 in the first innings before Australia notched 481 in reply with centuries from Hussey and Haddin.
