Defence Secretary John Healey said a frigate, planes and hundreds of personnel monitored a Russian attack sub and two spy submarines near undersea infrastructure north of the UK.
He said the Russian vessels eventually left after the operation that lasted more than a month.
Healey said his message to Russia was "we see your activity over our cables and our pipelines and you should know that any attempt to damage them will not be tolerated and will have serious consequences".
Representatives of Norway's defence and foreign ministries, as well as the armed forces, did not immediately respond to requests for comment.
British officials have tried to keep Russia in the international spotlight even as the world's attention is focused on conflict in the Middle East.
They have also stressed the overlap between conflicts there and in Ukraine, saying Russia has supplied Iran with drone parts and other support.
Healey told a news conference that "Putin would want us to be distracted by the Middle East", but Russia was the main threat to the UK and its allies.
"We will not take our eyes off Putin," he said on Thursday.
In late March, the UK said its military was ready to seize ships suspected to be part of Russia's "shadow fleet" of vessels shipping oil in violation of international sanctions over Moscow's war on Ukraine.
Previously, Britain had only helped France and the US monitor ships before they were boarded.
"We are ready to take action" against the vessels, Healey said.
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