The brief text adopted on Friday was drafted by Norway and Mexico.
"The Security Council expresses deep concern regarding the maintenance of peace and security of Ukraine," it reads.
"The Security Council recalls that all Member States have undertaken, under the Charter of the United Nations, the obligation to settle their international disputes by peaceful means."
"The Security Council expresses strong support for the efforts of the Secretary-General in the search for a peaceful solution," reads the statement, which also requests UN chief Antonio Guterres brief the council again "in due course".
Guterres welcomed the council support on Friday, saying he would "spare no effort to save lives, reduce suffering and find the path of peace".
Guterres met with Russian President Vladimir Putin in Moscow and Ukrainian President Volodymyr Zelenskiy in Kyiv last week.
His visits paved the way for joint United Nations and International Committee of the Red Cross operations that have allowed 500 civilians to leave Ukraine's port city of Mariupol and the besieged Azovstal steel plant in the past week.
The Security Council statement was agreed despite a diplomatic tit-for-tat that has been escalating since Russia launched what it calls a "special military operation" on February 24.
A council resolution needs nine votes in favour and no vetoes by the United States, Russia, China, France or the United Kingdom to pass.
The UN General Assembly, where no country has a veto, has previously overwhelmingly adopted two resolutions, illustrating Russia's international isolation over Ukraine.
The General Assembly deplored Russia's "aggression against Ukraine," demanding both that Russian troops stop fighting and withdraw and that there be aid access and civilian protection.
It also criticised Russia for creating a "dire" humanitarian situation.
Meanwhile US President Joe Biden has signed a new $150 million weapons package for Ukraine, providing additional artillery munitions, radar and other equipment in the latest in a series of transfers to help repel Russia's invasion.
"Today, the United States is continuing our strong support for the brave people of Ukraine as they defend their country against Russia's ongoing aggression," Biden said in a statement.
The US has rushed $3.4 billion worth of armaments to Ukraine since Russia invaded, including howitzers, anti-aircraft Stinger systems, anti-tank Javelin missiles, ammunition and recently-disclosed "Ghost" drones.
Biden warned the latest spending has "nearly exhausted" what he can authorise without further congressional approval
Ukraine said just 50 civilians were allowed to leave a bombed-out steelworks in the city of Mariupol on Friday, accusing Russia of violating a truce intended to allow all those trapped beneath the plant to depart after weeks under siege.
Mariupol has endured the most destructive bombardment of the 10-week-old war, and the sprawling Soviet-era Azovstal plant is the last part of the city - a strategic southern port on the Azov Sea - still in the hands of Ukrainian fighters.
Ukrainian officials say Russia has stepped up attacks on Azovstal in the hope of seizing it by Monday, when the Kremlin will mark Russia's victory over Nazi Germany in World War II.
UN-brokered evacuations of some of the hundreds of civilians who had taken shelter in a network of tunnels and bunkers beneath the plant began last weekend before being halted by renewed fighting.
The city's mayor has estimated 200 people remained trapped at the plant with little food or water.
Mariupol authorities earlier said Russian forces had fired at a car that was involved in an attempt to evacuate the plant, killing one Ukrainian fighter and wounding six.
Ukrainian President Volodymyr Zelenskiy described the blockade of Mariupol as torture and said if the Russian forces killed civilians or troops who could otherwise be released, his government could no longer hold peace talks with Russia.
