The drone attacked at dawn in the mountainous Tantil area, provincial government spokesman Abdul Ghani said.
The high-ranking Taliban leader killed in the attack was Maulvi Attaullah, successor to Haji Dawran Safi, who was killed in a similar attack in October, Afghan news agency AIP said.
However, the NATO-led International Security Assistance Force, or ISAF, in Afghanistan denied that it had carried out any operation in Kunar Tuesday.
Drone strikes are very controversial in Afghanistan, particularly when civilians are killed.
Last Friday a Taliban commander and an eight-year-old boy died in such an attack, leading Afghan President Hamid Karzai to threaten he would not sign a new military pact with the US if it continued such "cruelty" against his people.
The US and Afghanistan are negotiating a military pact that would maintain US troops on Afghan soil when NATO forces withdraw in 2014.
Although the Loya Jirga - the traditional assembly of elders - gave the go-ahead to the pact in November, Karzai now wants to postpone the signing until next year's general elections.
The US, however, wants to have it signed as soon as possible.
The rough draft of the pact contemplates the presence in the country of as many as 15,000 US soldiers up to 2024.