Well, if you are counting President Obama (who is half white), the current Supreme Leader of Iran, Ayatollah Ali Khamenei, is half Azeri (the dominant ethnic group in Iran is Persian).
King Abdullah of Jordan is also half white/british (majority ethnic group in Jordan is either Arab or Palestinian, depending whether or not you think Palestinians are distinct from Arabs), through his mother, Princess Muna al-Hussein (Birth name was Toni Gardiner).
Stalin was ethnically Georgian, and controlled the Russian-dominated Soviet Union for decades. He brought several other Caucasus people with him to the Kremlin; Lavrenti Beria (headed the NKVD; ethnically Mingrelian, a subgroup of Georgian) and Sergei Mikoyan (Chairman of the Presidium, de jure head of state of the USSR, from 1964-65; Armenian) are probably the most well-known and prominent.
First one that comes to mind is [Alberto Fujimori] (http://en.wikipedia.org/wiki/Alberto_Fujimori), who was president of Peru a while back.
[Deng Xiaoping] (http://en.wikipedia.org/wiki/Deng_Xiaoping) was Hakka amid the Han majority in Chinese politics.
[Hun Sen] (http://en.wikipedia.org/wiki/Hun_Sen) is the Sino-Khmer (ethnically Khmer and Teochew) Prime Minister of Cambodia.
The Shinawatras, [Thaksin] (http://en.wikipedia.org/wiki/Thaksin_Shinawatra) and [Yingluck] (http://en.wikipedia.org/wiki/Yingluck_Shinawatra) are both Sino-Thai (ethnically Thai and Hakka).
Off the top of my head:
The Assads in Syria are part of the Alawite religious minority; not technically an ethnic minority I suppose, but definitely a separate group within the country.
Belize has had a Prime Minister of Palestinian descent, Said Musa.
Two of the presidents of Ecuador, Abdala Bucaram and Jamil Mahuad, have come from the economically influential Lebanese minority there. While double-checking that, I found that Colombia, El Salvador, Honduras, Jamaica, the Dominican Republic and Argentina have all had Prime Ministers or Presidents of Arab descent.
This might not be what you're looking for, but if you go back far enough you can see a lot of countries being ruled by foreigners. England was ruled by the Normans after the Norman conquest, and the Normans were ethnically very different from the English at the time.
By that same token, there's the invasion of India by Muslim Turks in the early 13th-century. For the next few hundred years, India was ruled by ethnically Turkish Muslims when the majority of their subjects were Indo-Aryan Hindus.
In Asia:
Manmohan Singh, a Sikh Punjabi, was PM of India, whose majority population are Hindu and largest ethnic group are Hindi-speaking people.
BJ Habibie, 3rd president of Indonesia, is a Bugis, an ethnic group native of Sulawesi. The largest ethnic group in Indonesia is the Javanese, which comprised about 42% of Indonesian population.
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