One policeman and 10 civilians were wounded in the attack, hospital officials said.
Police spokesman Hazrat Hussain Mashrakiwal said the attack took place outside a police station, located near the palatial residence of the provincial Governor Attahullah Ludin in the provincial capital of Jalalabad.
The nearby state-run Radio and Television station was heavily damaged in the explosion.
A second bomb hidden in a motorized rickshaw and a third one in a vegetable cart, possibly detonated by remote control, also exploded near the police station, said the governor's spokesman, Ahmad Zia Abdulzai.
He told The Associated Press that two bodies could be seen lying outside the police station. Although Abdulzai said the situation was under control, witnesses in the area said gunfire was continuing and there were reports that insurgents had gained entry into the police station.
There were unconfirmed reports from the scene that some police had been killed.
Interior Ministry spokesman Sediq Sediqqi said in a Twitter message: "5 Terrorists killed by police in today's attack in Jalalabad, 1 or 2 of them still hiding in the building, the police operation is still on."
In a one-line Pashto-language email to news media, Taliban spokesman Zabihullah Mujahid said the attack was staged by Taliban insurgents.