Baku has warned it will "continue until the end" in the Armenian-majority region, over which it has fought two wars with neighbouring Armenia.
The latest flare-up cames as Russia, the traditional power broker in the region, is bogged down in a conflict in Ukraine, which has left it isolated by the West.
"We urge the conflicting parties to immediately stop the bloodshed, stop hostilities and eliminate civilian casualties," Russia's foreign ministry said in a statement.
It also said its 2,000-strong peacekeeping mission in Nagorno-Karabakh was evacuating civilians and providing medical assistance.
UN Secretary-General Antonio Guterres similarly demanded an end to hostilities.
He called in the "strongest terms for an immediate end to the fighting, de-escalation, and stricter observance of the 2020 ceasefire", said his spokesman Stephane Dujarric.
The United States and France rushed a day earlier to try to halt an Azerbaijani offensive.
Paris on Tuesday called for the UN Security Council to meet urgently on the crisis, which came just as world leaders gathered in New York for the annual General Assembly.
US Secretary of State Antony Blinken spoke by phone with the leaders of both Armenia and Azerbaijan, with French President Emmanuel Macron also speaking to Armenian Prime Minister Nikol Pashinyan.
Blinken urged Azerbaijan "to cease military actions in Nagorno-Karabakh immediately and deescalate the situation" in a call with President Ilham Aliyev, State Department spokesman Matthew Miller said.
Fears of a fresh war in the volatile Caucasus region have been growing recently, with Armenia accusing Azerbaijan of a troop build-up around the disputed territory.
Separatists said Azerbaijan on Tuesday pounded the mountainous territory with artillery, aircraft and drones.
Blasts rocked the separatist stronghold Stepanakert.
More than 7,000 people were evacuated from 16 villages, the separatists said.
Azerbaijan's defence ministry said it had taken control of more than 60 military positions during "localised anti-terrorist measures".
Armenian separatists said on social media that fighting was ongoing "along the entire line of contact" and that Azerbaijani forces were "trying to advance" into the territory.
Separatists said 27 people, including civilians, were killed and more than 200 wounded.
Baku said it would fight until the separatists surrendered.
"Illegal Armenian armed forces must raise the white flag," Azerbaijan's presidency said.
"Otherwise, the anti-terrorist measures will continue until the end."
The ex-Soviet Caucasus rivals fought two wars over the mountainous territory, in the 1990s and in 2020.
The latest offensive raised fears that the unrest could destabilise the region.
As angry protesters clashed with police in Armenia's capital Yerevan, calling on Pashinyan to resign, the country's security council warned of large-scale unrest, vowing to take "effective measures" to maintain constitutional order.
More than 30 people were injured in the clashes, the health ministry said.
Accusing Azerbaijan of engaging in "ethnic cleansing of Karabakh Armenians", Pashinyan said in a televised address that the Armenian army was not involved in the fighting and the situation on the border between Armenia and Azerbaijan was "stable".
Armenia's foreign ministry called on Russian peacekeepers to "take clear and unequivocal steps to stop" the fighting.
Azerbaijan justified its operation citing "systematic" shelling by Armenian-backed forces and accusing them of carrying out "reconnaissance activities" and fortifying defensive positions, accusing separatists of "a high level of combat readiness".
Russia and Turkey, which oversee a fragile peacekeeping mission in Nagorno-Karabakh, had been informed about the operation, Baku said.
Turkey, a historic ally of predominantly Muslim Azerbaijan that views mostly Christian Armenia as one of its main regional rivals, called the operation "justified", while urging "comprehensive negotiations".
Armenian separatists blamed international inaction for the fresh fighting.
"By ignoring warnings about Azerbaijan's criminal intentions and refusing to act accordingly, all the responsible international actors failed to prevent yet another Azerbaijani (act of) aggression" against Nagorno-Karabakh, they said in a statement.
The fighting erupted hours after Azerbaijan said four police officers and two civilians were killed in mine blasts in Nagorno-Karabakh, with authorities blaming separatists.
Earlier, Armenian separatists said they had reached an agreement with Azerbaijani authorities to resume aid deliveries to Karabakh.
When the Soviet Union collapsed in 1991, ethnic Armenian separatists in Karabakh broke away from Azerbaijan.
The ensuing conflict claimed some 30,000 lives.
A six-week war in 2020 saw Armenia cede swathes of territory it had controlled since the 1990s.