During his visit to the blast site, Energy Minister Berat Albayrak confirmed the casualty toll and said one of the injured was in "serious condition".
Earlier, speaking at the 7th Turkey Energy Summit that began in Adana Thursday, Albayrak pointed out the hand of "puppeteers" in the attack.
"You are mistaken if you think that you will scare and discourage this nation with traditional methods or distract it," he said, adding the attacks would not constrain Turkey from going ahead with its direction.
Before the energy minister's remarks, Adana Governor Mahmut Demirtas told Anadolu Agency the car had exploded at 8.05 a.m. [0605 GMT] in the parking lot close to the office entrance.
“It is suspected that a female carried out the attack,” Demirtas said, adding the license plate of the suspect's car had been identified.
Later, police seized a bomb-laden car and a suspect in the Sarihamzali neighborhood of Adana's Seyhan district after the morning attack.
The police shot the suspected driver as he attempted to flee, an anonymous police source said.
Several ambulances and police teams had rushed towards the area after the bomb attack, while a police helicopter also hovered above.
Nearby vehicles and the governor's building were also damaged in the incident.
Police have launched an investigation into the incident. A man, who worked at the parking lot, was taken to a police station for questioning, a police source, who spoke on condition of anonymity due to restrictions on talking to the media, said.
Meanwhile, the Prime Ministry imposed a temporary broadcast ban on the blast, the Supreme Board of Radio and Television announced.
An Adana court decided on a broadcast ban until the completion of police investigations.
Prime Minister Binali Yildirim was informed about the blast by ministers, according to a Prime Ministry source, who spoke on condition of anonymity due to restrictions on talking to the media.
About the incident, Deputy Prime Minister Numan Kurtulmus tweeted: "I condemn the treacherous attack in Adana, and curse terror and its supporters."
Kurtulmus said Turkey would "resolutely" continue to fight against terrorism and would not "surrender" to terrorism.
Foreign Minister Mevlut Cavusoglu also tweeted: "Terrorists and collaborators targeted our tranquility in Adana. I wish God’s mercy upon our citizens who lost their lives in this atrocious attack."
Turkey's EU Minister Omer Celik also tweeted: "Cursed terror continues to target our people. We will till the end struggle against this terror for humanity."
Celik wished a "speedy" recovery to the injured people. Later, Celik told reporters in Adana the perpetrators would be revealed "soon".
"The attack was perpetrated directly on the Governor’s office. It was seen that Governor [Mahmut Demirtas] was directly targeted," he added.
Parliament Speaker Ismail Kahraman also condemned the attack, saying: “I would like it to be known that these kinds of attacks which aim at damaging our country's security and people's tranquility, will never prompt even a slight retreat in our counter-terrorism determination."
Meanwhile, the main opposition Republican People’s Party (CHP) leader, Kemal Kilicdaroglu, called the governor and condemned the "treacherous" attack, according a written statement from the party.