Ukrainian Prime Minister Arseniy Yatsenyuk has urged countries not to lift sanctions against Russia until Kyiv regains control of all its territory, including Crimea.
Addressing the UN General Assembly, Yatsenyuk demanded that Russia pull back its forces from eastern Ukraine, "stop the supply of Russian-led terrorists" and start "real talks, peace talks."
Moscow denies sending troops into Ukraine.
Yatsenyuk said Russia must abide by "all points" of the cease-fire agreement reached between the rebels and Kyiv in Minsk on September 5.
He said that, despite the declared cease-fire in the Donetsk and Luhansk regions of eastern Ukraine, Ukrainian soldiers and civilians were dying from rebel shelling.
Yatsenyuk also delivered a defiant message to Vladimir Putin, telling the Russian leader that Moscow may win on the battlefield, "but you will never win the fight against the nation, the united Ukrainian nation."
The United States has imposed multiple rounds of economic sanctions targeting Russia's energy, defense, and financial sectors, as well as penalties on government officials and other individuals close to Russian President Vladimir Putin. The European Union has also ordered sanctions.
On the sidelines of the UN General Assembly, U.S. Secretary of State John Kerry held brief talks with Russian Foreign Minister Sergei Lavrov on the crisis in Ukraine.
According to the State Department, Kerry stressed the importance of quickly implementing all 12 points of the Minsk agreement.
Earlier in his address to the UN General Assembly, U.S. President Barack Obama said Russia’s annexation of Crimea in March harks back to an era "when large nations trampled small ones in pursuit of territorial ambition."
Reiterating Washington’s stance that Russia is fueling an armed separatist movement in eastern Ukraine and has moved its own troops across the border, Obama vowed to continue a push for international pressure on Moscow.
"We will impose a cost on Russia for aggression, and we will counter falsehoods with the truth," he said.
But the U.S. president also expressed hope that the cease-fire could bring a lasting peace if the Kremlin complies with it.
"If Russia takes that path -- a path that for stretches of the post-Cold War period resulted in prosperity for the Russian people -- then we will lift our sanctions and welcome Russia’s role in addressing common challenges," Obama said.
He added that the United States and its allies "will support the people of Ukraine as they develop their democracy and economy."
"We call upon others to join us on the right side of history -- for while small gains can be won at the barrel of a gun, they will ultimately be turned back if enough voices support the freedom of nations and peoples to make their own decisions," Obama said.
With reporting by AFP and AP