Authorities identified the gunman, who was firing a vintage carbine rifle, as an 18-year-old Austrian man but declined to comment on media reports that he was a known Islamist extremist.
German Chancellor Olaf Scholz said that police "may have prevented something terrible from happening today", declaring in a post on X that "anti-Semitism and Islamism have no place here".
German news outlet Spiegel and Austrian media reports said the gunman had been investigated last year for allegedly spreading Islamic State group propaganda, but the case was dropped.
Bavarian state premier Markus Soeder said "there is a terrible suspicion" the case was linked to Thursday's anniversary of the 1972 attack on Israeli athletes at the Olympic Games in Munich by Palestinian militants.
Eleven Israeli athletes and a German police officer were killed at the Games after gunmen from the Black September group broke into the Olympic village and took them hostage.
Thursday's shootout at around 9:00 am (0700 GMT) sparked a mass mobilisation of about 500 police in downtown Munich, where residents and office workers huddled indoors as sirens wailed and a helicopter flew overhead.
Video footage published by German media showed dramatic scenes in which police commandos in body armour and helmets took cover from gunshots, then unleashed a barrage of bullets.
Police said five police officers fired at the man, who died on the spot with his weapon beside him -- a rifle that was fitted with a bayonet, images showed.
German Interior Minister Nancy Faeser said, "The quick and decisive reaction of the Munich police stopped an attacker today and possibly prevented a terrorist act of violence".
"This act took place near the Israeli Consulate General –- on the anniversary of the terrorist attack on the Israeli Olympic team in 1972," she said.
Israeli President Isaac Herzog wrote on X that he had spoken with his German counterpart Frank-Walter Steinmeier.
"Together we expressed our shared condemnation and horror at the terror attack this morning near the Israeli consulate in Munich," he said.
"On the day our brothers and sisters in Munich were set to stand in remembrance of our brave athletes murdered by terrorists 52 years ago, a hate-fuelled terrorist came and once again sought to murder innocent people."
Herzog thanked the German security services for their "swift action" and said that "together we stand strong in the face of terror".
A memorial service for the victims of the hostage-taking in Fuerstenfeldbruck, where the Israeli athletes were shot, was cancelled, according to the Sueddeutsche Zeitung newspaper.
The Israeli consulate was unstaffed Thursday because of the planned remembrance event.
The exchange of gunfire sparked panic and a widespread police lockdown in a central area of the Bavarian state capital, near the Documentation Center for the History of National Socialism.
After securing the scene, Munich police wrote on X that there were "no indications of any other suspects" and that no one else had been wounded.
Soeder, the state premier, thanked police and said: "Munich held its breath for a time, there were moments of great fear about what could happen.
"Luckily it all turned out well in the end, no one was hurt and only the perpetrator was eliminated."
Since the start of the war between Israel and Hamas in Gaza after the October 7 attack, many Jewish communities worldwide have been targeted in attacks and hate crimes.
This is a special cause of concern in Germany, which in the aftermath of World War II and the Holocaust has committed itself to steadfast support for Israel.
A record number of 5,164 anti-Semitic crimes were recorded in 2023 in Germany, up from 2,641 the year before, according to domestic internal intelligence.
The Central Council of Jews in Germany estimates that there are around 100,000 practising Jews in the country and around 100 synagogues.
Fraser stressed that "it remains our duty to do everything we can to protect Jews in Germany and Jewish and Israeli institutions".