The blast led to the partial collapse of the five-floor building in Mansoura, 110km (70 miles) north of Cairo.
Interim Prime Minister Hazem Beblawi called it "an act of terrorism".
There have been a series of attacks on the security forces claimed by Islamist militants since the military deposed President Mohammed Morsi in July.
No group has so far said it was behind Tuesday's attack, but security officials said it bore the hallmarks of the Sinai-based Salafist-jihadist group, Ansar Beit al-Maqdis.
The military-backed interim government said it was an attempt to scare people ahead of next month's referendum on a new constitution - but that the violence would not disrupt voting.
Although Mr Beblawi did not directly blame the Brotherhood, one of his spokesmen said it had showed "its ugly face as a terrorist organisation, shedding blood and messing with Egypt's security".
At a funeral for the victims, who included at least 12 policemen, hundreds of people chanted: "The people want to execute the Brotherhood".
Some in the crowd held up posters reading "no to terrorist groups" and set fire to the car of a man who flashed a pro-Morsi symbol, while others attacked houses belonging to a leading member of the Brotherhood, according to the Associated Press.
The Brotherhood condemned the bombing, describing it as a "direct attack on the unity of the Egyptian people". It also accused the government of "exploiting" the violence to target its members and "create further violence, chaos and instability".
The explosion shook the building at about 01:10 (23:10 GMT on Monday).
Media reports say the provincial security chief was among those hurt. Officials said most of the casualties were police officers.
The blast shattered windows of nearby buildings and its impact was felt about 20km (12 miles) away, the reports say.
Mansoura - a city of about 480,000 people - is the capital of Dakahliya province in the Nile Delta region.
Since the removal of Mr Morsi - Egypt's first democratically elected president - his supporters have been staging mass rallies demanding his immediate release.
More than 2,000 Muslim Brotherhood members have been arrested, and 450 of them on Monday went on hunger strike in protest at their "inhumane treatment".
Mr Morsi is now facing three separate criminal trials relating to his time in office.
The first trial opened on 4 November, but has been adjourned until 8 January.
At least 15 people have been killed and more than 100 others injured in a car bomb attack on a security building in northern Egypt, officials say.