It said HMS St Albans monitored the Admiral Gorshkov's "activity in areas of national interest".
Russia has not commented on the issue.
The Royal Navy said there had been a recent "upsurge in Russian units transiting UK waters". Britain recently warned of a new threat posed by Russia to internet cables under the sea.
Air Chief Marshal Sir Stuart Peach, the chief of the defence staff, said earlier this month that Britain and Nato needed to prioritise protecting the lines of communication.He said it would "immediately and potentially catastrophically" hit the economy if they were cut or disrupted.
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In a statement, the Royal Navy said HMS St Albans was called upon to sail on 23 December and "keep watch on the new Russian warship Admiral Gorshkov as it passed close to UK territorial waters".
It said the British frigate remained at sea on Christmas Day, monitoring the Russian vessel. It would return to Portsmouth on Boxing Day.
"I will not hesitate in defending our waters or tolerate any form of aggression," Defence Secretary Gavin Williamson said.
"Britain will never be intimidated when it comes to protecting our country, our people, and our national interests."The Royal Navy said that HMS Tyne was also called to escort a Russian intelligence-gathering ship through the North Sea and the English Channel on Christmas Eve.
Meanwhile, a navy helicopter was scrambled to track two other Russian vessels.
Last January, a British warship and three RAF Typhoons escorted a Russian aircraft carrier and a number of other ships up the English Channel.
Relations between Britain and Russia have remained tense since Moscow's annexation of Ukraine's southern Crimea peninsula in 2014.