Atal told Ahmad Zaidan, Al Jazeera's correspondent in Islamabad, that the Taliban's spring attacks would not be affected by Dadullah's death.
He said that operations were continuing as planned, led by Mansour.
Mansour was freed in March as part of a prisoner swap for the release of Daniele Mastrogiacomo, an Italian journalist kidnapped while working in Afghanistsan.
Atal also said that the Taliban had arrested a close aide to Dadullah for treachery that had led to his killing.
He said: "We have captured the spy who helped US forces kill Mullah Dadullah. Each time there was a [coalition] strike, the man would disappear and then reappear after the bombing was over."
The Taliban have previously executed members accused of spying. Atal said the man had confessed to being a spy for the US.
In a separate development, the Taliban have claimed responsibility for an assassination attempt against Assaullah Khalid, the governor of Kandahar province.
Security sources in the city said that a suicide bomber drove a car laden with explosives into a column of cars that usually accompany Khalid.
The governor has said that his car was not within the column at the moment of the attack.