The attacker blew himself up outside the hall where thousands were gathering for a concert to mark Grozny City Day in the regional capital, an interior ministry spokesman told AFP.
He triggered the explosives after police securing the hall tried to inspect the man's identity papers, the spokesman added.
The Russian Investigative Committee said the bomber was suspected to be 19-year-old Opti Mudarov.
"Five police were killed and 12 others were injured," it said in a statement.
"At the cost of their lives, the police saved the lives of thousands of people," said Russian Interior Minister Vladimir Kolokoltsev, quoted by state news agency TASS.
Kadyrov said the bomber "wanted to ruin people's joy on their day of celebration, but he has not succeeded," according to the news agency Ria Novosti.
Following a failed uprising against the Russian government in 1994-1996, the Chechen separatist movement became increasingly radical, evolving into a violent Islamist insurgency in the 2000s.
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