Afghan officials said militants had carried out the attack in which four other people were killed.
At the same time, three suspected Taliban fighters were killed in a botched raid on a provincial leader's convoy.
The violence comes just days after Taliban commander Mullah Mohammed Omar purportedly threatened a major rebel resurgence.
Both of Saturday's attacks occurred in Ghazni, a province in eastern Afghanistan that has been spared the worst of the fighting in the past year and raised fears for its nascent future.
Ali Ahmad, a provincial police chief, said the former governor, Taj Mohammed Qari Baba, was ambushed as he was driving to his home in Ghazni city with a driver, bodyguard and two relatives. All were killed when their vehicle was riddled with gunfire.
Two men were arrested at the scene and were being questioned, Ahmad said.
Baba was a powerful pro-government figure in Ghazni and a critic of the Taliban.
Believed to be in his 60s, he was a commander in the mujahidin that fought occupying Soviet forces in the 1980s. He served twice as Ghazni's governor - between 1992 to 1995 and in 2002, after the Taliban was ousted.
Hours after Baba was killed, suspected Taliban rebels ambushed a convoy carrying Ghazni's current governor, Sher Alam, triggering a shootout that left three attackers dead. One of the assailants was wounded and arrested.
Alam said: "I am safe, and our security forces are looking for the Taliban, who had escaped toward a nearby village."
The attacks are the latest against prominent Afghans. Recent months have seen a string of political and religious leaders targeted.
A week ago, a car bomb in the capital, Kabul, hit a convoy carrying the speaker of the upper house of the Afghan parliament, injuring him and killing four people.
Rebel violence has increased in the past year. About 1600 people died in 2005, the most since US-led forces invaded the country and ousted the Taliban in 2001.