Both operations occurred Sunday, the Kuwait news agency KUNA reported.
The commander had been involved in attacks in Gulistan against Afghan security forces and coalition troops, military officials said. No one else was injured or killed in that operation.
The militants were reportedly killed about 20 miles from Kandahar.
The U.S. military lost a drone Sunday when it crashed shortly after launch in Paktika Province. Drones, pilotless aircraft, are being used for espionage and bombing of suspected terrorist targets in northwestern Pakistan.
Officials at Bagram Air Force Base said the drone had mechanical difficulties shortly after takeoff.