At least 46 people are dead and more than 70 injured by the blasts, which occurred just after lunch. At least 200 people are believed to have been in the headquarters and surrounding buildings at the time.
Chechen administration officials say about a ton of explosives carried in a truck and sport utility vehicle were set off by suicide bombers who sped through security check points around the government complex.
The Russian government is sending a mobile field hospital and medical teams to Grozny to aid in treating the injured.
A Chechen separatist web site reports that an unnamed Chechen commander says the attack was the work of Chechen suicide bombers.
Russian authorities immediately blamed the separatists for the explosions, but said the attack would not derail plans for a referendum on a new constitution for the war-torn republic.
The U-S State Department has strongly condemned the bombings and sent condolences to the families of the blast victims. In a statement, a department spokeswoman said the attack underscores that there is no military solution to the problems in Chechnya and called on Chechen leaders to break all ties with terrorist groups.
Russia has been trying to show that the situation in Chechnya is slowly returning to normal.
The leaders of the pro-Russian Chechen government were not in the building when the blasts occurred. Chechens who cooperate with Russian federal authorities are frequently targeted and killed by separatists.
Russian troops retreated from Chechnya in 1996 after a peace agreement with local authorities ended a two-year conflict. But they swept back in October, 1999, to put down a separatist insurgency in what Moscow has described as an anti-terrorist operation.