In the Taliban heartland in the south, an attack on a police checkpoint in Helmand province killed nine policemen. Also, rockets hit the grounds of the Kabul international airport but caused no damage.
Rockets also struck the NATO base at Bagram, just north of the Afghan capital, causing minor damage, the alliance said.
The Taliban offensive comes at a sensitive time this year, against the backdrop of the country's presidential election. Militants have also stepped up terror attacks to sow insecurity and weaken the government as international forces prepare to withdraw by the end of this year.
The attackers shot the guards, broke in and took over the building but it was unclear how many people may have been inside when the militants took it over.
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Afghan police and army forces arrived at the scene shortly afterward, surrounded the building and were engaging in sporadic exchanges of gunfire with the militants inside, said Gen Fazel Ahmad Sherzad, the provincial police chief of Nangarhar province.
Deputy Interior Minister Gen Mohammad Ayub Salangi said in a message posted to his Twitter account that the attack involved three militants, and that the police killed one of the attackers.
In the capital, Kabul, two rockets fired from unknown location landed inside the perimeter of the city's international airport without causing any damage, Kabul police spokesman Hashmat Stanikzai said.
To the north of the city, four rockets struck the Bagram military airbase, 48 kilometres from Kabul. The projectiles hit the Bagram Airfield around 3 am, resulting in minor damage to equipment and a building, said the international coalition.