Starting the fourth day of the Test 202 runs in arrears, England had reached 309-1 with a lead of 88 at the close of play with Cook on 132 not out and Jonathan Trott (54) at the crease.
Although they could not match the heights of Mike Hussey and Brad Haddin's partnership of 307 that put Australia in charge on the third day, Strauss and Cook's opening stand of 188 could end up being just as important at the end of the five-Test series.
Where touring sides of recent years might have crumbled, Strauss and Cook showed England's boasts of their teams' resilience was not just talk.
"It was very important that we came back into the game today," said England captain Strauss. "Yesterday was a very frustrating day for us and some of our batmen needed to score some runs today. Fortunately Alastair and I were able to." Strauss was removed for a duck in the third ball of the first innings but seized his chance in the second to claim his 19th Test century — his first since his 161 in the Lords Ashes Test last July - soon after lunch.
"It was my fourth Ashes hundred and every one of them is very special," said Strauss. "I had a bit of luck, it wasn't the best technical innings I've ever played, but in terms of importance it's up there." Cook had scored more slowly but was given fewer chances than his skipper and clinched his 14th Test century, only his second against Australia, with a cut shot that sent the ball fizzing to the boundary just after tea to add to his first innings 67.
By that stage England's best partnership in a Test at the Gabba had been broken and Strauss was furious with himself after coming down the wicket to occasional spinner Marcus North only to be stumped for 110 by wicketkeeper Haddin.
"It was not the sort of dismissal we were looking for," he said. "I was very disappointed with myself, both as a batsman and a captain." Cook was joined by Trott, who rode his luck a little with some loose shots but helped put on 121 for the second wicket as Australian heads dropped after a long, hard day in the field.
Questions will be asked about the Australian paceman, in particular Mitchell Johnson, who failed to get anything out of the Gabba strip and remains wicketless in the Test at the cost of 131 runs.
"We weren't able to sustain pressure for long enough," said Shane Watson, who thought it was unfair to single out Johnson.
"Unfortunately all of us as a bowling unit weren't able to execute for long enough to build up the pressure." Australia's misery was compounded by three missed catches, Johnson dropping Strauss when the England captain was on 70 and Peter Siddle and Michael Clarke spilling two more difficult chances to dismiss Trott.
Even the umpire referral system, which saved Hussey's innings early on Saturday, went against the hosts when umpire Aleem Dar's refusal of Ben Hilfenhaus's appeal for lbw against Trott was upheld after the review of the TV footage.
"It's ebbed and flowed," said Watson of the match. "And it's always going to be like that throughout the whole series. A couple of little passages will decide it and I hope there's a little passage in it for us tomorrow." Play was abandoned early because of bad light and will begin 15 minutes early at 09:45 local time (2345 GMT) on Monday.
