The bombing late Tuesday also injured 20 people, some of them critically, state-run RIA Novosti reports. The wounded include 16 police officers.
The BBC says it appears the suicide bombers used the first blast to draw police to the scene, and then allowed themselves to be arrested before setting off their bombs.
The second bombing came about 30 minutes after the first, when police and other emergency workers had arrived at the scene.
The bombings, which took place in a densely populated part of Grozny, near a parliament building, came as people in the mainly Muslim republic in Russia's North Caucasus celebrated Eid.
Russian prosecutors have released a statement identifying two of the attackers as young Chechen men, the BBC says. No militant group has claimed responsibility for the attack.
Investigators named the two men as Magomed Dashayev, 22, from the Chechen town of Urus-Martan and Adlan Khamidov, 21, a student in a village near Grozny, the Guardian says.
Chechnya's Kremlin-backed leader Ramzan Kadyrov said the attackers had chosen "the most sacred day for all Muslims" and "shown their real faces."
"Thousands of people are shedding tears instead of celebrating," a spokesman for Kadyrov said.
Kadyrov promised a harsh response to the bombings, calling the attackers "zombified bandits" and "not people, but the devil incarnate," the Guardian reports.
Rebels in Chechnya began fighting for independence in the 1990s, but in recent years the fight has become an increasingly radicalized Islamist insurgency, aimed at imposing Islamist rule, CNN says.