The minister, Thomas de Maiziere, also called for wider oversight for federal police as well as for a crisis management centre to be set up.
"We don't have federal jurisdiction to deal with national catastrophes. The jurisdiction for the fight against international terrorism is fragmented," he wrote in a guest column for the daily Frankfurter Allgemeine Zeitung.
"The federal police's scope of action is restricted to railway stations, airports and border controls," he wrote, stressing that "it is time" to reexamine Germany's security set-up.
Amri had been under surveillance since March, but German police dropped their watch in September, thinking that he was a small-time drug dealer.
The failed asylum seeker should also have been deported months ago but Tunisia had not provided the necessary paperwork until after the attack.
De Maiziere said federal departure centres should be set up to hold asylum seekers in the weeks or days leading up to their expulsion.
"The current remit of the federal police is too limited," he said.
"We need a set of common rules and better coordination, for instance in checking dangerous individuals," he said.
Further, the federal government should take charge of domestic intelligence services, said the minister.
De Maiziere is a close ally of Chancellor Angela Merkel, who is running for a fourth term in a general election expected in September.
Her government came under fire in the wake of the December 19 attack for its liberal border policy, which allowed in more than one million asylum seekers since 2015, and for allowing Amri to slip through the net despite documented security concerns.
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