"The secretary general is deeply concerned about continued violence in eastern Ukraine, which has led to loss of life, further instability and which is contributing to a climate of fear and anxiety," Xinhua quoted a spokesperson of Ban as saying in a statement.
"As the stakes are now so high, the secretary-general is seriously concerned that the situation could quickly spin out of control with consequences we cannot predict."
"He stresses, in the strongest terms, the necessity for all parties to honour their commitments under the Geneva Statement," the statement said, referring to an accord reached in Geneva earlier this month by top diplomats from Ukraine, Russia, the US and the European Union on the crisis of Ukraine.
The four-way talks reached an agreement last on Thursday, calling for an immediate halt of violence in Ukraine and practical steps to de-escalate the tension.
"Military action must be avoided at all costs," said the UN statement. "The secretary general calls on all sides to immediately refrain from violence, intimidation or provocative actions and find a way forward toward de-escalation."
At least five people were killed and one wounded on Thursday during an "anti-terror" operation in an eastern Ukrainian city, while Russia's top diplomat said the US should use its influence to make Kiev fulfill its commitment to the Geneva agreement.
During "counter-terror" actions at three checkpoints set by a pro-Russia militia outside the city of Slavyansk, five "separatist militants" were killed, the Ukrainian interior ministry said in a statement.
A new wave of unrest erupted in eastern Ukraine early in April, when pro-Moscow activists seized government buildings in the cities of Donetsk, Lugansk and Kharkov.
Kiev has repeatedly blamed Moscow for inciting the unrest and splitting Ukraine. Russia denies the charges.