Bukich was the starting quarterback in his senior year at Roosevelt High School in St Louis, Missouri, on a team that did not win a game. He played one game as wingback when Roosevelt tried to upset eventual league champions Cleveland by surprising Cleveland by playing the single wing.
Bukich earned a football scholarship to Iowa State University as a wingback. He then transferred to the University of Southern California. During his senior year, Southern California went to the 1953 Rose Bowl. After the starting quarterback (an All-American) was sidelined with injuries, Bukich came in and conducted a drive, completing all but two of his passes, that resulted in the only score, for USC to defeat Wisconsin, 7-0. Bukich was selected as the Most Valuable Player in the game. After his career, he was inducted into the USC Trojan Hall of Fame and, later, into the Rose Bowl Hall of Fame.