Oleksandr Turchynov said the rebels used a Russian-made anti-aircraft system, and a senior general was among the dead.
The town of Sloviansk has seen fierce fighting between separatists and government forces in recent weeks.
President-elect Petro Poroshenko has vowed to tackle “bandits” in the east.
The helicopter was hit during heavy fighting between Sloviansk and Kramatorsk, reportedly after it had dropped off troops at a military base.
Oleksandr Turchynov said the 14 dead included Gen. Serhiy Kulchytskiy, head of combat and special training for Ukraine’s National Guard.
It is one of the worst losses of life for government forces in the conflict so far. Last week at least 14 soldiers died in a rebel attack on an army checkpoint near Donetsk, some 80 miles from Sloviansk.
Earlier this month the separatists shot down two army helicopters, also near Sloviansk, killing a pilot and another serviceman.
Petro Poroshenko, a confectionery magnate, won 54.7% of the vote in last Sunday’s presidential election, according to final results announced on Thursday.
After the poll, he called the separatists “terrorists” intent on maintaining a “bandit state”. He vowed to tackle them “in hours”, not months.
The conflict has intensified in recent days. The rebels say they lost up to 100 fighters when they tried to seize Donetsk airport on Monday.
Sloviansk has long been the centre of heavy fighting. Pro-Russia militiamen seized four international monitors there on Monday.
The four – a Dane, an Estonian, a Turk and a Swiss national – are members of the Organization for Security and Co-operation in Europe (OSCE).
The self-proclaimed mayor of Sloviansk, Vyacheslav Ponomaryov, told Russia’s Interfax news agency they were safe and well and could be released soon.
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