Bombings are rare events in Kazakhstan. The country, with a Muslim majority, prides itself on being the most stable nation in Central Asia and is seen as attractive by foreign investors.
News reports said security police were searching for accomplices and had cordoned off a block of the city of 277,000 inhabitants.
According to local news-site Tengiz News, the bomber was taking revenge for recent arrests in Kazakhstan of Muslims who are followers of the fundamentalist Sunni branch of Islam known as Wahhabism.
A regional Muslim leader condemned the suicide bombing.
Until now, Kazakhstan has been spared violence, unlike neighbouring Uzbekistan, Tajikistan and Kyrgyzstan.
Its president, Nursultan Nazarbayev, hosts conferences on tolerance of world religions on a regular basis.