After first playing football Tanasije Kuvalja decided to take up cycling and soon became one of the best Yugoslav cyclists of his day. In 1968 Kuvalja was selected to compete at the Mexico City Olympics, where he placed 16th in the team time trial with his compatriots. One year later he became the Yugoslav national road race champion, a title he would win four more times during his career. He had further success in the early 1970s with multiple top 10 finishes in stages at both the Tour of Algeria and the Peace Race. Kuvalja retired from cycling when he was 29-years-old but later became a successful motor racing driver, winning two editions of the Serbia Touring Car Championship.