Ukraine officials said there was very little left to defend in the bombed-out eastern city, where hundreds of civilians remain trapped in a chemical plant.
The order to withdraw on Friday came four months to the day since Russian President Vladimir Putin sent tens of thousands of troops over the border, unleashing a conflict that has killed thousands, uprooted millions and reduced whole cities to rubble.
Luhansk regional governor Serhiy Gaidai said troops in Sievierodonetsk had already received the order to move to new positions.
"Remaining in positions smashed to pieces over many months just for the sake of staying there does not make sense," Gaidai said on Ukrainian television.
The withdrawal will mark the biggest reversal for Ukraine since the loss of the strategic southern port of Mariupol in May.
The latest Russian advances appeared to bring the Kremlin closer to taking full control of Luhansk, one of Moscow's stated war objectives, and set the stage for Sievierodonetsk's twin city of Lysychansk to become the next main focus of fighting.
Vitaly Kiselev, an official in the Interior Ministry of the separatist Luhansk People's Republic - recognised only by Moscow - told Russia's TASS news agency it would take another week-and-a-half to secure full control of Lysychansk.
Russia invaded Ukraine on February 24, but abandoned an early advance on the capital Kyiv in the face of fierce resistance bolstered by Western arms.
Since then Moscow and its proxies have focused on the south and Donbas, an eastern territory made up of Luhansk and its neighbour Donetsk, deploying overwhelming artillery in some of the heaviest ground fighting in Europe since World War II.
Ukraine on Friday again pressed for more arms, with its top general, Valeriy Zaluzhniy, telling his US counterpart in a phone call Kyiv needed "fire parity" with Moscow to stabilise the situation in Luhansk.
South of Sievierodonetsk, Ukrainian soldiers also withdrew from the towns of Hirske and Zolote in the face of overwhelming Russian forces, said Oleksiy Arestovych, an adviser to Ukrainian President Volodymyr Zelenskiy.
Arestovych said the orderly retreat from the towns was a good thing in that it broke with a Soviet and post-Soviet military tradition to never retreat, whatever the circumstances.
He said Ukraine's military had learned the hard lesson of trying to defend positions at all cost during battles with pro-Russian forces in 2014.
Russian troops had entered Hirske and fully occupied the surrounding district on Friday, municipal head Oleksiy Babchenko said.
"There is a red flag flying over the municipal administration (in Hirske)," a official told Reuters by telephone.
Ukraine's foreign minister played down the significance of the possible loss of more territory in the Donbas.
"Putin wanted to occupy the Donbas by May 9. We are (there) on June 24 and still fighting. Retreating from a few battles does not mean losing the war," Dmytro Kuleba said in an interview with the Italian newspaper Corriere della Sera.
The general staff of Ukraine's armed forces said its troops had some success in the southern Kherson region, forcing the Russians back from defensive positions near the village of Olhine, the latest of several Ukrainian counter-assaults.
Ukrainian media showed footage of a school smouldering and gutted by Russian shelling in Avdiivka - a town in Donetsk region just inside Ukrainian-held territory.Â
The war has had a massive impact on the global economy and European security arrangements, driving up gas, oil and food prices, pushing the EU to reduce its heavy reliance on Russian energy and prompting Finland and Sweden to seek NATO membership.
In a major sign of support, European Union leaders this week approved Ukraine's formal candidature to join the bloc - a decision Russia said on Friday amounted to the EU "enslaving" neighbouring countries.
