The youngest of five children from an Armenian father and English mother, he also claims Scottish ancestry through his maternal grandfather, a Sanderson, and the MacDonell of Glengarry clan. Oundjian was educated in England, where he began studying the violin at age seven with Manoug Parikian. He attended Charterhouse School in Godalming. He continued studies later with Béla Katona. He then attended the Royal College of Music.
Oundjian studied at the Juilliard School with Ivan Galamian, Itzhak Perlman, and Dorothy Delay. While at Juilliard, he minored in conducting, and later received encouragement his endeavors when he attended a master class from the eminent German conductor Herbert von Karajan.
In 1980, Oundjian won First Prize at the International Violin Competition in Viña del Mar, Chile. Oundjian became the first violinist of the Tokyo String Quartet and held the post for 14 years. A repetitive stress injury forced Oundjian to curtail his instrumental career. He then shifted his full-time musical focus to conducting.