The man, identified by Palestinians as Mohammed al-Shaham, 21, was "neutralised" and died in an Israeli military hospital, police said.
The man's father, Ibrahim al-Shaham, told AFP that his son was shot in the head at point-blank range and was left bleeding for 40 minutes.
According to al-Shaham, police knocked on their door at 3:30 am and "we didn't manage to make it to the door, Mohammed got there first, and the door exploded".
"They started shooting inside the house, the first shot was in Mohammed's head," he said, adding that the officers left him bleeding before arresting him and taking him away.
Jerusalem Border Police launched the raid to locate weapons in the town of Kafr Aqab, which is part of Israeli-annexed east Jerusalem but located on the West Bank side of the security barrier.
Al-Shaham said that one of the officers had told him that they got the wrong house.
Police denied those claims to AFP, saying "this was the right house and this was the suspect," although no weapons were seized.
The Palestine Liberation Organisation's new secretary general, Hussein al-Sheikh, called for "an immediate and urgent international investigation" into what he labelled a "criminal execution".
On Sunday, eight Israeli and US citizens were wounded when a Palestinian gunman opened fire on a bus just outside Jerusalem's Old City.
That shooting came a week after the end of a deadly three-day conflict between Israel and Islamic Jihad militants in the Palestinian coastal enclave of Gaza.