The intense fighting, which the U.N. says has killed more than 5,350 people since April, comes ahead of a crucial summit involving Western leaders on Wednesday, as well as peace talks later Tuesday.
Ukrainian President Petro Poroshenko told Parliament that Russian-backed rebels launched an artillery strike on the town of Kramatorsk, which is more than 50 kilometers (30 miles) away from the front line.
Poroshenko said the first round of rocket fire hit the headquarters of the Ukrainian command in eastern Ukraine and the second landed in a residential area.
The government-controlled Donetsk regional administration said three people were killed and 15 injured in the artillery strike.
Kramatorsk was the site of major fighting until July when pro-Russian separatists retreated from it.
Local website Donetskiye Novosti posted photos from the scene, showing an artillery shell stuck in the ground next to a residential building and two bodies lying nearby.
The volunteer Azov battalion, loyal to Kiev, said on social media on Tuesday that it captured several villages northeast of the strategic port of Mariupol, pushing the rebels closer to the border with Russia.
However, rebel military spokesman Eduard Basurin said in a televised news conference that the rebels have not retreated.
The Azov said rebels shelled the village of Kominternove, east of Mariupol, causing unspecified civilian casualties.
An Associated Press reporter at a government check-point between there and government-controlled Mariupol was told of ongoing fighting several miles away.
Two ambulances and four pick-up trucks carrying Ukrainian troops were seen coming from the direction of Kominternove toward Mariupol.
The rebels reported advances, too.
Basurin said late Monday that they have surrounded the railway hub of Debaltseve, the focus of fierce fighting in the past weeks, cutting it off from a major highway.
A video posted online by a rebel-sympathizing website showed the separatists moving along the highway while the bloodied bodies of Ukrainian soldiers lay on the side of the road.
At least seven Ukrainian troops were killed overnight in the east, Ukrainian military spokesman Anatoliy Matyukhin said on Tuesday.
In the rebel stronghold of Donetsk, which comes under constant shelling, two civilians were killed and 12 injured.
The fierce fighting and a mounting death toll comes amid renewed efforts to work out a peaceful solution in a conflict that has displaced at least 1 million people and left the powerhouse of the Ukrainian economy in ruins.
Representatives of Ukraine, Russia, rebels and the Organization for Security and Cooperation in Europe are meeting for long-anticipated talks later on Tuesday to lay the groundwork for a meeting on Wednesday between the leaders of Russia, Ukraine, Germany and France.
The Kremlin on Tuesday warned the West ahead of the talks against sending weapons to Ukraine or putting pressure on Russia.
Russian President Vladimir Putin's spokesman, Dmitry Peskov, told the Russian News Service radio station on Tuesday that any talk about imposing new sanctions on Russia or arming the Ukraine government would destabilize the situation in Ukraine.
The talks in Minsk on Tuesday were tentatively scheduled for 6 p.m. local time (1600 GMT), the Belarusian foreign ministry said.
In Washington on Monday, President Barack Obama and German Chancellor Angela Merkel rallied behind efforts to reach a long-shot diplomatic resolution to the conflict, but they offered no clear path for how the West would proceed if talks this week fail.
Merkel staunchly opposed arming Ukraine's beleaguered military while Obama dangled the prospect that the U.S. could for the first time send weapons to Ukraine.