No one was injured in the September attacks that hit the city, which is the birthplace of the anti-immigration and Islamophobic PEGIDA movement.
Investigators raiding two sites secured items that could be used to build explosives, police said, adding that forensic tests show that "the DNA traces secured are consistent with the DNA of the suspect".
Bild daily identified the suspect as Nino K. and said he had spoken at a PEGIDA rally in the summer of 2015, railing against "criminal foreigners" and "lazy Africans".
The attacks occurred just days before Dresden was due to host national celebrations to mark 26 years since the reunification of East and West Germany.
The first homemade bomb damaged the door of the mosque while the imam and his family were inside. The second blast struck at the main venue for the German reunification anniversary festivities.
Dresden, a Baroque city in Germany's ex-communist east, has become a hotspot for far-right protests and hate crimes after more than a million asylum seekers arrived in Europe's biggest economy since 2015.
In an annual report outlining progress since reunification, the government warned in September that growing xenophobia and right-wing extremism could threaten peace in eastern Germany.
Correction: This article originally stated that the suspect was 29. New information from prosecutors has changed this to 30.