On Friday dozens of people died in a fire at a building taken over by pro-Russians in Odessa in the south-west.
The US condemned the violence as “senseless”.
Mr Avakov said Ukrainian forces had on Saturday taken a television tower in Kramatorsk, which is about 17km (10 miles) from the pro-Russian separatists’ stronghold of Sloviansk.
“We are not stopping,” Mr Avakov wrote on his Facebook page.
Footage shows people trying to escape from the building in Odessa She says the defence ministry also reported “serious fighting” last night in Andreevka, some 6km west of Sloviansk.
The ministry said pro-Russian separatists had attacked the armed forces and two Ukrainian soldiers were killed. Pro-Russian groups said there were civilian casualties.
The operations come a day after running battles between pro-Kiev groups and separatists in Odessa, which culminated in the fire at the Trade Unions House.
Officials said some people were overwhelmed by smoke and others died after they jumped from the building.
Trade Unions House in Odessa was engulfed with flames after rival groups exchanged petrol bombs
Pro-Russian crowds took to the streets of Odessa, with some armed men threatening Kiev supporters
The violent clashes followed a rally by Ukrainian football fans in the south-western port city The exact sequence of events is unclear, but reports suggest separatists had barricaded themselves inside the building and both sides were throwing petrol bombs.
Local police chief Petro Lutsiuk told Reuters news agency on Saturday that more than 130 people had been arrested following the violence.
The US said the unrest was “unacceptable” and urged all sides to work together to restore law and order in light of the “tragedy”.
Ukrainian Deputy Foreign Minister Danylo Lubkivsky told the BBC that an investigation was under way but insisted Russia was behind the violence.
Earlier on Friday four people were killed in clashes between rival groups in Odessa – the first such eruption of violence in the south after weeks of mounting unrest in the east.
The death toll in Odessa is the highest since violence broke out between pro-EU protesters and police in the capital in February.
The Black Sea city is ethnically mixed with a large number of Russian speakers.
‘A criminal adventure’
Pro-Russian gunmen have seized scores of government buildings in eastern Ukraine in recent weeks and have detained several international observers.
Fergal Keane reports from outside Sloviansk where violent clashes have claimed several lives On Friday President Olexander Turchynov said that Ukrainian armed forces had killed many separatists in an offensive in Sloviansk.
Government forces took over pro-Russian checkpoints outside the city but separatists shot down two Ukrainian army helicopters, killing a pilot and another serviceman.
The US called for commitments made in Geneva last month to be implemented, referring to the agreement signed by all sides to dissolve illegal militia operating in eastern Ukraine.
The West continues to accuse Moscow of provoking the unrest in eastern Ukraine, something Russia denies.
Eastern Ukraine has a large Russian-speaking population. It was a stronghold for President Viktor Yanukovych before he was overthrown by pro-Western protesters in February.
Russia then annexed the Crimean peninsula – part of Ukraine but with a Russian-speaking majority – in a move that provoked international outrage. (BBC)