Wogan said: "I've had 35 wonderful years commenting on the Eurovision for radio and television. From my first in a music hall in Dublin to my last in the huge arena at Belgrade, it has been nothing but laughter and fun.
"The silly songs, the spectacle, the grandiose foolishness of it all. I'll be sad to leave it all behind, but I'll be watching and listening, and happy to leave it in the witty hands of Graham."
Wogan had already cast doubt on whether he would cover next year's competition claiming that it was "no longer a music contest" due to bloc voting among friendly neighbouring countries.
He lashed out when former 'X Factor' finalist Andy Abraham only received 14 points at this year's contest in May, which was won by Russia with 272 points.
Wogan first provided commentary for the Eurovision Song Contest in 1973 and again in 1978, before taking on the commentary role completely from 1980.
Norton, who will host the competition from next year when it will be held in Moscow, described it as "an amazing job and a huge honour".
He said: "Sir Terry is nothing less than legend and is an impossible act to follow but somebody must and I just couldn't say no.
"I can't wait to get to Moscow -- with a combination of cheap vodka and a language barrier what could possibly go wrong?"
Norton will host 'Your Country Needs You' on BBC One in the New Year, giving the UK public the chance to choose the act that will perform Andrew Lloyd Webber's specially composed song in Moscow.
Norton's association with the Eurovision title began with the 'Eurovision Dance Contest' which he has presented for the past two years.