He was born on March 3, 1928, in the village of Aksengir. He graduated from the acting department of the Almaty Theater and Arts College in 1950.

In 1949, he made his film debut with the role of the Chukchi Ayye in the movie Alitet Leaves for the Mountains, directed by Mark Donskoy. Donskoy had also selected other future prominent actors, including Nurmukhan Zhanturin (who played the role of Tumatughe) and Gulfayrus Ismailova.

From 1950 to 1967, he performed on the stage of the Kazakh Theater for Young Audiences (Kazakh TЮZ). From 1958, he was an actor at the Kazakhfilm studio, where a documentary film about him, The Return of Kenenbay, was made in 1975. In the late 1960s, during a fight with bandits, he sustained a severe injury and became disabled.

For the rest of his life, Kozhabekov was confined to a wheelchair, but he continued to act, mainly in episodic roles and those where walking was not required. The film The Wolf's Pit highlighted his acting skills, even though he was confined to a wheelchair.