However, the BBC star admitted that was one “weird” condition she had to follow to before he signed it off – cut her trademark fringe.
Discussing her role in Strictly, she told the crowd at the Edinburgh TV Festival: “I don’t say this in a faux self-deprecating way. It is a machine. A steam engine. I’m not even a cog.
“Danny Cohen gave me the job and phoned me up and said, ‘We’d love you to do it.’ I said, ‘Thank you so much, that is such an honour.’
“He added: ‘I’m just going to need a phone number from you’ and I was like, ‘Oh yeah, OK, which one?’ and he said, ‘Your hairdresser’s’.
“I was like, ‘Bit weird, controller of BBC One’.”
Winkleman continued: “Two days later, James, who does my hair, said, ‘This guy called Danny phoned up to complain about the length of your fringe.’
“He (Danny) was like: ‘You can have this job but it’s just annoying, it’s down to your nose. You want to be on telly but you’re invisible’.”
Elsewhere, the presenter admitted she didn’t feel concerned about the show being fronted by two women when she officially started in 2014.
She said during her chat at the TV festival: “I felt incredibly nervous at the time but the two women thing, the biggest show on television by miles at that point, ratings-wise, in recent history was Bake Off and that was hosted by two women [Mel Giedroyc and Sue Perkins].
“So I was, like, I’m not even pulling on that string.”
Winkleman received the Outstanding Achievement Award at Thursday’s festival in honour of her three decade TV career.
Aside from her Strictly role, she has enjoyed recent successes with The Piano on Channel 4 and BBC One’s The Traitors, which are both getting new runs this autumn.