Horn was born in San Francisco, California, His mother is a pediatric oncologist from Croatia, his father is an eye surgeon, and he has a younger brother, Philip. Horn speaks some Spanish, is fluent in Croatian, and has studied Mandarin.
In 2010, Horn won on Jeopardy!, during Kids Week, earning $31,800 after wagering $12,000 during Final Jeopardy! Producer Scott Rudin was among those viewers impressed by Horn, subsequently offering him an audition for the role of Oskar Schell in Extremely Loud and Incredibly Close. Horn's performance in the movie was praised by critics "for its sensitivity and emotional depth".
Since EL&IC and Space Warriors, Horn has returned to being a student at Piedmont High School. He's a strong student but also plays classical piano and is active in dance. He still auditions and wants to pursue acting further but he is still sticking with his studies in the case of a backup plan.