Separatists said Ukrainian forces killed three of their fighters and two civilians when they moved in on Sloviansk in the early hours. Kiev said two helicopter crew had died and seven servicemen had been wounded in the operation.
On the other side of Ukraine, police said three people were shot dead and 15 others wounded in clashes between people backing Kiev and pro-Russian activists in largely Russian-speaking Odessa.
The southern Black Sea port lies close to Crimea, which was annexed by Moscow in March following the overthrow of Ukraine's pro-Moscow president by protesters who opposed his decision to scrap a trade deal with Europe.
President Vladimir Putin's spokesman said Ukrainian forces had fired on civilians from the air in Sloviansk in a "punitive operation" that destroyed an international peace plan. Moscow has tens of thousands of troops on the border and claims the right to invade if needed to protect Russian-speakers.
The Western-backed government in Kiev said the use of missiles that brought down its helicopters was evidence Russian forces were in the town. Moscow denies its troops are on the ground.
Ukraine's acting president also said Russian "armed saboteurs" had tried to enter the country overnight, but were pushed back by Ukrainian border troops. He gave no further details. Russia's Security Service said the report was untrue.
Kiev said Moscow was backing groups in eastern Ukraine who were "putting civilians in danger, seizing hostages and creating an atmosphere of terror and violence".
Reporters in Sloviansk, the most heavily fortified bastion of pro-Russian separatists in eastern Ukraine, heard shooting break out and saw one helicopter opening fire before dawn. Later in the day the city was largely quiet, with shops shut and armed separatists in control of the streets.
Advancing Ukrainian forces in armoured vehicles took up positions closer in to the suburbs, but rebels still controlled most of the town of 130,000.
Acting President Oleksander Turchinov said the operation had been complicated by the rebels' use of human shields.
The growing chaos is overshadowing a presidential election the leadership in Kiev is planning for May 25. The rebels are planning a vote on May 11 to seek a mandate to break with Kiev, like one held in Crimea before Moscow took it over.
The United States and Europe have imposed sanctions on individuals over the Ukraine crisis but they have had limited impact. US President Barack Obama said the next step would be sectoral sanctions and they would be imposed if Russia impeded the Ukrainian presidential poll.
On the square outside city hall in Sloviansk, about 100 people gathered on Friday and said they were appealing to Mr Putin to send troops to help them.
On the town's southern outskirts, eight Ukrainian armoured personnel carriers cut off the road but faced a cordon of local residents shouting at them to go home. Some rebels erected barricades of trees.
Mr Putin's popularity has soared with the seizure of Crimea and talk of restoring Moscow's former empire. This week he restored the Soviet-era tradition of holding a May Day parade on Red Square, where marchers carried banners hailing the acquisition of Ukrainian territory.
The European Union said it was watching events in eastern Ukraine with growing concern. But Kiev is not a member of Nato and Western leaders have made clear they will not fight to defend Ukraine.