The resolution gave the UN the authority to administer Kosovo, which has been a UN protectorate since the NATO bombing of the former Yugoslavia ended a conflict between Albanian and Serb forces in 1999.
The 2,000-strong police, justice and civil administration EU mission has already been launched, and is expected to be fully deployed by June. A 17,000 NATO force is to remain in Kosovo.
"The so-called European Union 'Rule of Law' mission was launched without the required decision of the Security Council," Vitaly Churkin said at an emergency session of the UN Security Council.
"The parameters of the European Union Mission in Kosovo, above all from the viewpoint of the mandate given to it in Brussels, did not agree with the provisions of resolution 1244," he added.
He added that the EU mission, aimed at helping and training Kosovan police, judges and penitentiary system workers, also contradicted the mandate on an international presence in Kosovo.
Churkin also reiterated Russia's position that the Kosovo problem could only be solved with the Security Council in a "leading role."
A UN Security Council meeting on Kosovo's unilateral declaration of independence ended late on Monday without a resolution or joint statement.
Serbian President Boris Tadic said that Kosovo's actions represented a precedent which would cause "irreparable damage to international order."
In further developments, Belgrade has recalled its ambassador from the U.S. over Washington's recognition of Kosovo.
Besides the U.S., France, the U.K., Germany, Australia, Turkey and Italy have all so far recognized Kosovo. Poland is expected to follow suit on Tuesday.
Spain, with its own separatist problems, announced on Monday that it would not acknowledge the sovereignty of the former Serbian province. Three other EU states, Cyprus, Romania and Slovakia, earlier announced that they would also refuse to recognize Kosovo.
Russia remains deeply opposed to Sunday's unilateral declaration of independence by Kosovo, with Foreign Minister Sergei Lavrov telling the U.S. on Tuesday that the move threatened international stability.
