Police earlier said they were checking witness statements that the alleged assailant, a 27-year-old man who was under arrest and the only suspect, shouted "Allahu Akbar" (God is Greatest in Arabic).
But Bavarian Interior Minister Joachim Herrmann said he had received no indications that the man had a migrant background, telling reporters: "So far there are no findings that are relevant for state security."
He said the man had shown up in a different part of Germany a few days ago, suspected of using drugs and exhibiting behaviour suggesting he might be mentally disturbed.
Bavarian radio said he used a knife with a 10 cm (4 inch) blade in the apparently indiscriminate attack, which took place around 5 a.m. (0300 GMT) in Grafing, a commuter town about 32 km (20 miles) southeast of the Bavarian capital in southern Germany.
One victim, a 50-year-old, died of stab wounds in hospital shortly after the attack. Police said three other men aged between 43 and 58 were injured, with Bavarian radio saying one was seriously hurt.
Germany, which is playing a supporting role in the fight against Islamic State, has not suffered a major attack by Islamist militants on the scale of those that have hit neighbouring France and Belgium, Reuters reported.
But with about 260 of the more than 800 home-grown radicals who have joined jihadist groups in Syria and Iraq having since returned to Germany, ministers have warned an attack is possible and security services are on alert.