The daytime gun battle came hours after Russian forces staged a pre-dawn raid, killing four fighters and seizing two portable anti-aircraft missile systems from rebels near the main air base.
The fighting in and around the Chechen capital was some of the fiercest in several months.
Russian television showed helicopters circling over a nine-storey apartment building in Grozny as heavily armed soldiers fired repeatedly.
Smoke later could be seen billowing out of the building, which, like many buildings in the Chechen capital, is already ruined from nearly a decade of war and violence in the southern Russian region.
A spokesman for Chechnya's regional interior ministry said at least five fighters and at least two federal forces were killed in the 30-minute battle.
An emergency situation ministry official said special forces had blocked off the building after being tipped off that rebel fighter Doku Umarov there. The emergency ministry said six rebels were killed along with three federal agents.
Russia's Itar-Tass and Interfax news agencies said six fighters were killed. Itar-Tass said Umarov - a Chechen rebel leader who is wanted by Russia forces for various alleged attacks - was not among the killed militants.
Grozny local television reported four civilians were wounded and at least one person died of a heart attack.
Major General Ilya Shabalkin, chief spokesman for the federal forces in the North Caucasus region, said the early morning raid took place in a village near the Khankala air base.
Chechnya has been gripped by
warfare for almost a decade
The four dead suspected militants included a citizen of the Central Asian nation of Kazakhstan, he said.
"The bandits under cover of darkness were sneaking around and looking for a vantage point to fire the Strela anti-aircraft missile to shoot down aircraft," Shabalkin said in a statement.
Russian forces left Chechnya in 1996 after a disastrous, 20-month war with separatists. Fighting resumed in 1999, when Chechnya-based insurgents made raids into a neighbouring region and after a series of deadly apartment-house bombings in Russian cities that officials blamed on the rebels.
Nearly three-quarters of the Chechen population is out of
work and electricity and telephone services are largely
Tens of thousands of people have fled, mostly to neighbouring regions and violence has spread to other Caucasus regions.
Rampant abductions of civilians - committed by separatist fighters, Russian forces and Kremlin-allied paramilitaries - have destabilised the region, leaving the populace fearful.