"This defeat should not have happened, and I accept responsibility. I'm here to face the consequences," Stojkovic told a press conference after the match played in the southern town of Leskovac.
"I spoke with the president and secretary general of the Serbian Football Association.
"I have offered my resignation, they will make an announcement also, but I will not travel to Andorra and lead the team," he said, quoted by the state-run RTS broadcaster.
He referred to Serbia's next World Cup qualifier against Andorra on Tuesday.
Saturday's result was "really poor, I did not expect this", the 60-year-old said.
"I have also spoken to the players and told them everything. It's not a tragedy when a coach leaves, and Serbia will always have a great supporter in me. I have led my country more than 50 times."
The former midfielder, nicknamed Piksi, took over as Serbia coach in 2021.
As a player he made 84 appearances for the former Yugoslavia between 1983 and 2001, and was one of the first top-tier European players to venture into Japanese football.
Stojkovic spent several years at J-League side Nagoya Grampus Eight, between 1994 to 2001.
After retiring as a player, Stojkovic was briefly chairman of Red Star Belgrade, before turning to coaching first Nagoya, and later Chinese side Guangzhou.
Serbia are third in their group K, with England on top and Albania second.
Meanwhile, Serbian media reported that Stojkovic could be replaced by former player and football manager Veljko Paunovic.
Saturday's match has been moved from the capital Belgrade to Belgrade upon Serbia's request to UEFA over security fears.