Ralf Jaeger accused Cologne police of making "serious mistakes".
More than 500 criminal complaints have been filed - 40% alleging sexual assault - relating to 31 December.
The Pope said Europe's "humanistic spirit" risked being undermined.
The "immense influx" of migrants was causing problems, he said, but the continent had the means to strike a balance between protecting its citizens and helping migrants.
Mr Jaeger, the state interior minister of North Rhine-Westphalia (NRW), says that recent arrivals in Germany are among the suspects in the attacks, which took place in central Cologne, in the area of the cathedral and the main railway station.
Nineteen individuals are currently under investigation by the state police in connection with the attacks, NRW's interior ministry says in a report (in German), none of them German nationals.
Those 19 suspects include 14 men from Morocco and Algeria. Ten of the suspects are asylum seekers, nine of whom arrived in Germany after September 2015.
The other nine are possibly in Germany illegally, the interior ministry says.
On Friday it emerged that of the 31 suspects identified by federal police - responsible for the station itself - 18 are asylum seekers, including nine Algerians and eight Moroccans.
The scale of the assaults on women in Cologne and other German cities has shocked Germany.
Mr Jaeger is himself under political pressure. On Friday he fired Cologne's police chief, Wolfgang Albers.
Addressing a committee of state MPs on Monday, Mr Jaeger criticised the police for not calling for reinforcements on the night, and also for the way they informed the public about the investigation in the days after the events.
His report details how a group of around 1,000 men of North African and Arabic origin gathered in the area on the evening of 31 December.
From this large group, smaller groups of men formed, who then surrounded women and threatened and attacked them, he said.
These smaller groups were predominately made up of North African men who had travelled to Cologne from different cities.
"After the intoxication with drugs and alcohol came violence," said Mr Jaeger. "It culminated in the acting out of fantasies of sexual omnipotence.
"That must be severely punished."
Around 1.1 million asylum seekers arrived in Germany in 2015.
Chancellor Angela Merkel's immigration policy has come under criticism since the attacks.
Riot police used water cannon to disperse anti-migrant protesters in Cologne on Saturday.
On Sunday evening six Pakistanis and a Syrian man came under attack in the centre of the west German city.
The attacks on Sunday took place in the early evening. In the first, a group of around 20 people attacked six Pakistanis, two of whom had to be treated in hospital.
Separately, five people injured a Syrian man who did not need hospital treatment.
Monday's report into the attacks in Cologne says that the combination of group sexual violence with robbery had not previously been seen in Germany. It notes that similar crimes took place in other parts of Germany on the 31 December, including in Hamburg.
The report describes a modus operandi known as "taharrush gamea" in Arabic, meaning group sexual harassment in crowds, and compares it to incidents reported in Cairo's Tahrir Square at the time of the Egyptian revolution.
A joint federal and state working group has been set up to examine the phenomenon and how to combat it.