The 15-member council threatened to impose sanctions on supporters of the Democratic Forces for the Liberation of Rwanda (FDLR) who have ignored a January 2 deadline to surrender.
The appeal from the council came a day after UN Secretary General Ban Ki-moon spoke by phone with Kabila to urge him to take "decisive action" against the FDLR rebels.
The UN's 20,000-strong MONUSCO force is preparing operations to drive out the rebels, but the mission's success hinges on an active role by the Congolese government troops.
"The Security Council calls on the authorities of the DRC and in particular, President Kabila as commander in chief, to approve swiftly and implement fully the MONUSCO-FARDC joint directive," said a council statement.
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The United Nations is pushing for the disarming of dozens of rebel and splinter groups after two decades of conflict in the eastern DR Congo, much of it fueled by the lucrative trade in minerals.
The council said in its unanimous statement that operations against the FDLR must start "immediately."
African regional leaders are due to discuss the planned offensive at a summit in Luanda on January 15-16.