The militants also set off a car bomb outside the wall around the airport, causing civilian casualties, said the military.
The dead and wounded civilians from the attack came from neighborhoods located near the airport, said Umar Ayub, a local hospital official. The 36 wounded included six women and three children, and several people were in critical condition, said Ayub.
Five militants were killed in a gunbattle with security forces during the attack, said the military.
However, Zahid Khan, a police explosives expert, said it appeared that three of the militants were accidentally killed when the car bomb they used to try to break through the airport wall exploded.
Four of the attackers who were killed were wearing suicide vests that have been defused, said the air force.
The air force side of the airport was the target of the attack and authorities were searching the area for any remaining attackers, said Defense Minister Naveed Qamar.
No air force personnel were wounded in the attack and none of the service's equipment was damaged, the air force said.
Local TV footage showed people in the neighborhoods near the airport rushing for safety as the attack occurred. One car was damaged in the attack and another was set on fire. A house was also damaged.
The airport was closed, and flights were diverted to other cities, said Pervez George, a spokesman for the Civil Aviation Authority.
No group has claimed responsibility for the attack, but suspicion will likely fall on the Pakistani Taliban. The militant group has been waging a bloody insurgency against the government for the past several years and has attacked Peshawar many times in the past.
Also Saturday, police said a judge freed a couple on bail who confessed to killing their 15-year-old daughter in October by pouring acid on her after their other children pardoned them.
The girl's parents, Mohammad Zafar and his wife Zaheen, said in a televised interview that they killed her because she sullied the family's honor by looking at a boy.
They were freed from a jail in Pakistan-administered Kashmir on Friday after their other children, who are minors, said they forgave their parents, said police officer Tahir Ayub. The children spoke through their guardian, who is also a relative, said Ayub.
The police officer said authorities had evidence to prove the murder charge against the parents, but by law, their children had the right to forgive them. The murder charge will likely be dropped, said Ayub.
The girl's death underlined the problem of so-called "honor killings" in Pakistan, where women are often killed for marrying or having relationships not approved by their families or because they are perceived to have somehow dishonored their family.
According to the Human Rights Commission of Pakistan, at least 943 women were killed in the name of honor last year. The real toll is believed to be higher because many of the crimes go unreported.