There was no immediate claim of responsibility but security sources said police raided the home of a suspected Islamic State militant believed to have carried out the attack and detained his father for DNA tests and questioning.
Turkey has suffered attacks recently both by Kurdish militants and members of Islamic State, raising concern at home and among NATO allies about the increasing spillover of conflict from neighbouring Syria. The city of Gaziantep is just 65 km (40 miles) from the Syrian border.
The Gaziantep-based suspect is believed to have detonated a bomb-laden vehicle just outside the gates of the city's main police headquarters on a street housing several other provincial government buildings, whose windows were also smashed.
"The father of a suspect who is believed to have carried out the attack has been detained. We have records of the suspect's links with Islamic State," a security source said.
Sources also said there were two vehicles carrying out the attack. While the suicide bomber was inside the detonated car, three men opened fire on police guarding the station from a second.
Several security sources also said police had received intelligence on the attack on Saturday and had ordered officers not to gather in front of the station as they deployed for May Day celebrations, a move which may have prevented a higher toll.
Nineteen police and four civilians were wounded, a statement from the governor's office said. One officer died at the scene and a second later in hospital, a security source said.
"We have eight people in intensive care. Seven of them are police officers," Deputy Prime Minister Mehmet Simsek told reporters after visiting the wounded.
Multiple threats
Several hundred miles east in the town of Nusaybin, three Turkish soldiers were killed and 14 wounded in an armed attack by Kurdish militants, an army statement said. Another was killed in clashes with PKK militants in border province of Sirnak.
A separate car bomb attack carried out by suspected PKK militants on a gendarmerie station in the southeastern town of Dicle wounded 10 troops, security sources said. Two others were wounded when an armoured vehicle en route to the scene was also attacked.
Turkey is facing security threats on several fronts. As part of a U.S.-led coalition, it is fighting Islamic State in Syria and Iraq and battling Kurdish PKK militants in its southeast, where a 2-1/2-year ceasefire collapsed last July.
Turkish military sources said on Sunday drones from the U.S.-led coalition, drawing on intelligence from Ankara, had struck an Islamic State explosives depot in the northern Syrian town of Dabiq. Two Islamic State militants outside the building were killed and several others were thought to have been inside.
Gaziantep, which borders Islamic State-held Syrian territory, is home to a large Syrian refugee population and there have been several police raids on suspected Islamic State militants there over the past months.
A wave of suicide bombings this year, including two in Turkey's largest city Istanbul, have been blamed on Islamic State, and two in the capital Ankara were claimed by a Kurdish militant group.
The Kurdish militant group TAK, an offshoot of PKK, on Sunday claimed a suicide bombing last week in Turkey's fourth-largest city of Bursa that wounded eight people.