"We killed him, he was our target... all such officers who are active against us will suffer the same fate," Azam Tariq, a spokesman for the group, said. "He was conducting different operations against us ... one of our [suicide bombers] has done this job."
The Frontier Constabulary assists other police and military units, and has been involved in a number of operations against Taliban fighters over the last few years.
"We have lost a very brave and able official in this cowardly attack," said Mian Iftikhar Hussain, the information minister for Khyber Pakhtunkhwa province, of which Peshawar is the capital.
At least 10 people, mostly civilians, were also wounded in the explosion, Al Jazeera's Imran Khan reported.
Pakistan's Express 24/7 television channel broadcast CCTV footage of the explosion, which shows Ghayur's car waiting in traffic at an intersection when the bomb went off.
Police officials in Peshawar have said the bomber was likely on foot, and that he waited for Ghayur's car to approach the intersection.
Peshawar has been the site of many deadly attacks over the last few years, but this is the first high-profile attack in several months.
The last flare-up of violence in Peshawar came in April, when a suicide bomber struck a market, killing more than 25 people. That bombing came two weeks after gunmen attacked the US consulate in the city.