Blasts at mosque and convention centre overshadow events marking the 10th anniversary of the German Islam Conference.
"Although no one has so far claimed responsibility, we must assume that there was a xenophobic motive," Horst Kretschmar, the Dresden police chief, said.
German Interior Minister Thomas de Maiziere said the mosque attack was "all the more scandalous" because it happened on the eve of the 10th annual meeting of the German Islam Conference.
Police linked the explosion at the congress centre to celebrations due to take place next week in Dresden marking the 26th anniversary of German unification, which is to be attended by German President Joachim Gauck.
"We have now switched to crisis mode," Kretschmar said, as police deployed to guard the city’s two mosques and an Islamic cultural centre.
About 300 worshippers regularly attend Friday prayers at the Fatih Camii mosque, which lies a short distance from Dresden’s historic centre.
The explosion at the mosque was detonated at 1953 GMT on Monday. The force of the blast pushed the front door of the building inwards and left the building covered with soot, police said.
The explosion at the convention centre - about 2 kilometres from the Fatih Camii mosque on the River Elbe, which runs through Dresden - occurred about half an hour later.
The heat caused by the explosion at the centre destroyed the side of a decorative glass cube in an open area in the congress building and resulted in parts of the building being evacuated.
Dresden, a Baroque city in Germany’s ex-communist east, is also the birthplace of the anti-immigration Pegida street movement, short for Patriotic Europeans Against the Islamisation of the Occident.
Its members have angrily protested against the influx of refugees and migrants that last year brought one million asylum seekers to Europe’s biggest economy.
About a dozen demonstrations are planned over the weekend, by both Pegida and by anti-fascist groups.
Saxony state premier Stanislaw Tillich called the "cowardly" bombings an "attack on freedom of religion and on the values of an enlightened society" that could easily have claimed lives.
Far-right hate crimes targeting shelters for asylum seekers in Saxony rose to 106 in 2015, with another 50 recorded in the first half of this year.
In an annual report outlining progress since reunification, the government warned last week that growing xenophobia and right-wing "extremism" could threaten peace in eastern Germany.
De Maiziere said he understood that many Muslims in Germany did not wish to apologise for every act of "terrorism" that is carried out in the name of Islam.
But he said he expected more from the Muslim organisations in Germany.
"I think it would be advisable that the security debate becomes more intense and also more public in the future," he said.
"Political influence from abroad in Germany through religion is something we cannot accept," said de Maiziere.
However, Muslim leaders attending the 10th anniversary of the dialogue forum hit back.
It was wrong, "to brand Muslims as representatives of foreign powers and to speak of them as having a representive role", said Bekir Alboga, the secretary-general of the Turkish-Islamic Union for Religious Affairs.
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