The 27-year-old, who had spent a stint in a psychiatric facility, had intended to target a music festival in the city of Ansbach but was turned away because he did not have a ticket.
Syrian refugee arrested after killing woman with machete in Germany
"Unfortunately, this is a terrible new attack which will surely increase people's anxiety," regional interior minister Joachim Herrmann said, adding that investigators "have not ruled out" he had an Islamist motive.
Hermann said he was worried "the right to asylum would be undermined" by the events of the past week, which has seen attacks on a train and shopping mall in the southern German state.
Sunday's explosion happened just outside a bar in Ansbach city centre, not far from where more than 2,000 people had gathered for the concert, at around 10:00 pm (2000 GMT).
Police have blocked off the area and emergency services were at the scene. Bomb experts were also sent to determine the cause of the blast.
Ansbach deputy police chief Roman Fertinger said there were "indications" pieces of metal had been added to the explosive device.
The perpetrator was killed in the explosion, police said in a statement, and a spokeswoman said 12 people were wounded, three of them seriously.
German military training 100 Syrian migrants in pilot project
Herrmann said the attacker, who came to Germany two years ago but had his asylum claim rejected after a year, had tried to kill himself twice in the past and had spent time in a psychiatric clinic.
Europe has been on edge for months after a string of deadly attacks claimed by the Islamic State jihadist group, including bombings in Brussels and carnage at Bastille Day celebrations in the southern French city of Nice.
In Bavaria, which has been a gateway for thousands of Syrian refugees, nine died in a shooting rampage in Munich on Friday and several people were wounded in an axe attack on a train near Wuerzburg.
Police released more details of Munich attacker David Ali Sonboly on Sunday, saying the 18-year-old was depressed and had spent two months in a psychiatric unit last year.
The teen was obsessed with mass killings and spent a year preparing for the gun attack that killed nine people, most of them foreigners.
At least 35 people were also wounded during Sonboly's shooting spree, which began at a McDonald's branch and ended with him turning his 9mm Glock pistol on himself.
Investigators have ruled out any link with the jihadists, though he appears to have planned the assault with chilling precision.
Police have also arrested a 16-year-old Afghan friend in connection with the attack.
Hundreds of people, many of them in tears, gathered outside the Munich shopping centre where the attack took place to pay tribute to the victims on Sunday.
Already steeped in grief and shock, Germans were further rattled by news that a Syrian refugee had killed 45-year-old Polish woman with a machete in the city of Reutlingen.
Munich gunman planned shooting for a year, chose victims randomly
Police said that incident on Sunday, in which three others were injured, did not bear the hallmarks of a "terrorist attack".
"When a man and woman have an argument, we assume that we are dealing with a crime of passion," a local police spokeswoman told DPA.
Three people were also injured in the attack, which ended when the 21-year-old assailant was hit by a BMW.
NTV showed amateur video footage of the suspect running away from the scene before cutting to him lying on the ground, his face bloodied and his hands cuffed by police.
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