Militants barricaded themselves in a school and police cordoned off city streets in violence that erupted just as violence has continued in Chechnya and nearby regions, with a suicide blast rocking Chechnya in October.
The Caucasus Emirate, a group that has claimed responsibility for a string of deadly attacks, said it was behind the violence in a video posted on the website Kavkaz Center, saying it was revenge for "oppression of Muslim women."
Chechnya's head Ramzan Kadyrov said nine militants were killed after several hours of fighting in the city, with armed personnel carriers dispatched to residential areas and shooting in the streets.
The operation against the gunmen killed 10 police and injured 28, Russia's National Antiterrorism Committee said in a statement.
It said police efforts "liquidated" the group and "thwarted major acts of terror" while some security actions were ongoing.
Pictures from the city showed considerable damage with debris and bullet castings littering the streets and a central market destroyed by fire.
Kadyrov said the militants had planned to stage the attack on Friday but had launched their violent action after traffic police stopped them in the early hours of Thursday.
Militants stormed a media centre called the Press House in central Grozny as well as a school about a kilometre away.
Kadyrov said on Echo of Moscow radio that the gunmen were "very heavily armed", including with grenade launchers.
An AFP correspondent said a large area in the city centre was closed off by security forces, with automatic fire as well as larger calibre artillery audible several blocks away.
Authorities said gunmen inside the Press House were killed, as it stood smouldering and gutted from fire.
Speaking in the Kremlin, Putin said he was certain that "local police, will suitably manage" to bring down the "latest raid by the terrorists."
Chechnya's parliament speaker Dukuvakha Abdurakhmanov blamed Western leaders for the attack, as Moscow is locked in the worst standoff with Europe and the United States since the Cold War over the Ukraine conflict.
"The dreams of Obama, Merkel and their conspirators will never come true," he vowed to TASS news agency.
Second attack in weeks
Thursday's attack was the first such raid in months by militants in Grozny, which was ravaged by two wars between separatists and the Russian army over the past 20 years.
But it comes weeks after five policemen were killed and a dozen wounded in Grozny in October when they stopped a young suicide bomber from attacking a concert hall where thousands had gathered to mark a local holiday.
That attack shattered a period of relative calm in the region and sparked concerns of a new cycle of violence in the North Caucasus.
"they have shown once again that they exist and continue doing what they were doing before," said Alexander Cherkasov of Memorial human rights group.
"There is an insurgency in Chechnya, despite careful and total control over the republic ,"he added.