Born in Pusto Šilovo, Socialist Federal Republic of Yugoslavia, he started his youth career in FK Partizan as a nine-year-old, although minimum age required to start playing for Partizan youths was ten. He was spotted by Partizan coach Virgil Popescu who insisted in bringing him over. Apart from short spell abroad, Blagoje spent most of his time in Partizan and, as a player or coach, he was part of Partizan for almost half a century. Paunović played for ten seasons with FK Partizan, amassing nearly 300 official appearances with the Belgrade club. During the 70's, he formed an efficient backline alongside Dragan Holcer and Miroslav Pavlović.
On the international level, Paunović played a total of 39 matches for Yugoslavia, making his debut on 12 November 1967 in a 4–0 home win against Albania for the UEFA Euro 1968qualifiers, and participating at the final stages in Italy.
Paunović's son, Veljko, was also a footballer and later manager of Chicago Fire, Reading, and the Serbian U18, U19, and U20 teams. He too started his extensive career with Partizan, but played mainly in Spain.