Thats a hard question, the way you've posed it, because 'culture' doesn't really indicate a clean homogenous group that rules another, and there'll be few examples because most leaders of nations could be said to be quite disconnected from the winning ruled people's culture anyway, and the kings or emperors were the conquerers really, not the people. But i'll have a go at providing a cleaner example for the sake of the challenge.

The 'Mongols' Kitan or Liao, when they defeated 'China', they knew they wouldn't be able to hold their rule over the coastal populations without having a seat of power. So outwardly they adopted the practices of the Chinese. This is apparently why the Forbidden City was initially Forbidden. Because the mongols used that area, among others, to continue to live privately in tents like they'd always lived.

https://www.birmingham.ac.uk/research/perspective/the-forbidden-city.aspx

Another less clean example would be the 1066 Norman takeover of Britain. The Normans undoubtedly left their mark, but they never actually erradicated the anglo-saxons culture. And in doing so the two cultures mixed. You can see artifacts of this in the etymology of english words, for instance the difference between 'beef and cow' i think is a good example.