As well as increasing security at Herstedvester, near Copenhagen, the ministry announced inspections at prisons across the country and the recruitment of several specialists to reinforce the regime at other high-security jails.
The changes came after Madsen, jailed for life for the 2017 murder of Swedish journalist Kim Wall when she went to interview him on board his homemade submarine, briefly broke out of Herstedvester Prison on October 20th.
During the break-out, 49-year-old Madsen used a fake explosives belt to threaten a prison psychologist.
Although he was only free for a few minutes before police surrounded him, the fact he managed to get that far meant that security was not good enough, said a ministry statement.
Prison officials had not been sufficiently aware of Madsen's willingness to plan and carry out serious crimes to make his escape, said the statement.
Madsen had been allowed unsupervised access to a workshop and, unaccompanied by staff, had been able to smuggle out the fake devices he used for his escape, said Justice Minister Nick Hækkerup.
A prison service report into the incident said Madsen had been allowed to move about inside the prison with a backpack without being challenged by guards.
Security has been stepped up in several blocks of the prison. Prisoners now have to carry all their personal effects in transparent bags and there is increased video-surveillance.
Since his escape bid, Madsen and five other prisoners have been transferred to another prison, the location of which has not been made public.