Writing in the Times newspaper on Friday, the Team Sky boss said the culture he promoted was one of "winning, but not winning at all costs".
"We were not sexist, but we were definitely 'medallist'.
"That is why we pushed for equal number of male and female events so our elite female athletes could have the same maximum chance of success as their male counterparts. It is why British Cycling has won as many female medals as male ones since 2008."
Cooke, the 2008 Olympic road race champion, last month told a British Parliamentary committee investigating doping in sport that cycling was a sport "run by men, for men".
The 33-year-old retired rider agreed when asked whether she felt sexism was culturally embedded in British cycling.
British Cycling and Team Sky have been under scrutiny over a mysterious package delivered to the 2011 Criterium du Dauphine in which Bradley Wiggins, the 2012 Tour de France winner, was riding.
The package was transported to France by British Cycling women’s team manager Simon Cope, who has said he did not know what was in it.
Cooke told the committee that Cope should have been concentrating on his job with the women.
"The facts are, they did nothing for the women. Whilst a deluxe program ran out for the men’s London 2012 bid, Emma Pooley and myself self-funded our flights to and accommodation in Australia,” she said then.
Brailsford said that reading some of the headlines of recent weeks might have given the impression that Britain's Olympic and Paralympic success was "forged in a boot camp for bullies and sexists".
While recognising that every sport can always do better, he hoped there could also be a balanced debate about what it takes to win at the elite level.
"Without putting too fine a point on it, 20 years ago Britain was largely an Olympic nation of gallant losers. Today we take Olympic and Paralympic success for granted," he wrote.
"I was, and still am, an advocate of UK Sport's 'no compromise' ethos. It is a ruthless approach and one that is completely meritocratic."
(Reporting by Alan Baldwin; editing by Mark Heinrich)
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