Stojkovic stepped down after Serbia's 1-0 loss to Albania in a World Cup qualifier that left their direct qualification hopes in tatters, with the team sitting third in Group K behind England and Albania.
The Serbian Football Association moved quickly to secure Paunovic, who guided the country's Under-20 team to World Cup glory in 2015, as they scramble to salvage their campaign with two matches remaining.
Paunovic led Oviedo's promotion push as they returned to LaLiga after a gap of 24 years, but the club sacked the 48-year-old this month as they struggled in the top flight and won two games to sit 17th when he left.
"Veljko Paunovic has shown from day one that he puts the interests of the national football team above all else," Branko Radujko, the Secretary General of the Football Association of Serbia, said in a statement.
"His priority was not a contract or financial terms, but to immediately make himself available to the Football Association of Serbia and take over the team at such an important moment."
With England having already topped the group, Serbia can no longer qualify directly but remain in the hunt for a playoff berth.
Paunovic faces the immediate challenge of matches against England on November 13 and Latvia three days later.
"The results of these two matches will in no way affect Veljko Paunovic's status in the future," Radujko added.
"We will continue discussions about his long-term contract after the end of these matches, regardless of their outcome."