Jaukovic is an ex-solider decorated by a former Serbian president.
It said the attacker threw the explosive device "into the United States embassy compound" from an intersection near a sports centre.
Police say they are still investigating Jaukovic's motives and whether he had acted alone.
"There was an incident", a police official, who declined to give his name, told Reuters.
Photos posted on Jaukovic's Facebook profile include a plaque honouring his contribution in the fight against North Atlantic Treaty Organisation during the bombing against Serbian military and police targets to stop genocide against Albanians in Kosovo. There was no immediate official confirmation of his military past or awards. In May 2017, Jaukavic wrote on the social media site "no to North Atlantic Treaty Organisation".
NATO's eastward expansion, from its onetime Cold War boundaries into the nations that once were part of the Soviet Union, is a huge frustration for Russian Federation.
The small Balkan country is in the process of tightening cooperation with the U.S.as it became the newest North Atlantic Treaty Organisation ally previous year.
Local media reports have stated that the police have confirmed two explosive devices were detonated near the premises. Eyewitnesses reported seeing a man throw an object, believed to be a hand-grenade, over the wall of the embassy that then exploded. The blast killed the man but did not injure anyone in the embassy, which was closed for the night.
Interior Minister Mevludin Nuhodzic said that the ministry and government "strongly condemn the attack on the U.S. Embassy", expressing regret and saying he authorities were committed to "securing the full safety and security" of embassy staff.
The U.S. Embassy has canceled all visa services for Thursday and access will be on an emergency basis only.
The attack came five minutes before midnight on Wednesday, web portal www.portalanalitika.me said, adding that citizens in the area heard two explosions.
Warning U.S. citizens of an "active security situation", the embassy cautioned them to avoid the embassy site and take precautions.
"Avoid large gatherings and demonstrations, and follow the instructions of local authorities", the note from US officials continues.
"Of course, violence is not something that we would condone, but we are waiting for the results of this investigation", Kocijancic said.
There was no immediate indication of any link.
The Balkan country declared independence on June 2006, after it separated itself from the Federal Republic of Yugoslavia, a federation it established with Serbia in 1992.