"I hope this conflict will be resolved, that they can leave the territory where they are unhindered," Putin said during a visit to Minsk, the capital of Belarus. "But the participants in this process should draw the necessary conclusions from this."
The demonstrators who overran the building in Luhansk are seeking at the very least a referendum on granting greater authority to Ukraine's regions.
Eastern Ukraine, which has a large Russian-speaking population, was the heartland of support for Viktor Yanukovych, the ousted president who fled to Russia in February. The government that replaced him in Kiev has resisted those demands so far, fearing they could lead to a breakup of the country or mean that more regions could join Russia, as Crimea did.
The storming came as 1,000 demonstrators gathered in front of the building. About 150 people, some masked and wielding baseball bats, broke out of the crowd and charged into the building, meeting no resistance. Later protesters formed a corridor to allow police inside the building to leave.
Luhansk, a city of about 450,000, is just 15 miles west of the border with Russia.
Regional autonomy is a core issue in the turmoil in eastern Ukraine, where insurgents fear that the government in Kiev will suppress the Russian-speaking population.
In Kiev, Ukraine's parliament on Tuesday discussed the possibility of holding a national referendum on whether the country should remain united or become a loose federation that allows the regions more powers. But no consensus was reached on how such a referendum would be phrased or when it could be held. Ukraine is already holding a presidential election May 25.
Yulia Tymoshenko, a former prime minister who is running for president and whose party dominates the new government, spoke out Tuesday against excessive decentralization.
"While we are giving authority to local administrative bodies, we are obliged under any circumstances not to lose authority over the country," Tymoshenko told lawmakers.
Russia has massed tens of thousands of troops near the Ukrainian border, feeding concerns that Moscow aims to use turmoil in the east as a pretext for an invasion.