The 190-strong company aboard the HMS St Albans set sail on 23 December to monitor the Admiral Gorshkov frigate and tracked its progress through what the Royal Navy called areas of national interest.
The Navy said there had been a recent "upsurge" in Russian units travelling through UK waters, including on Christmas Eve, when HMS Tyne was called to escort a Russian intelligence-gathering ship through the North Sea and the English Channel.
A Wildcat helicopter from 815 Naval Air Squadron, based at RNAS Yeovilton, was also scrambled to track two other Russian vessels.
HMS St Albans will return to Portsmouth on Boxing Day.
The Defence Secretary, Gavin Williamson, said: "I will not hesitate in defending our waters or tolerate any form of aggression.
"Britain will never be intimidated when it comes to protecting our country, our people and our national interests."
Relations between Britain and Russia are strained, and the UK Foreign Minister Boris Johnson said there was "abundant evidence" of Moscow meddling in foreign elections during a trip to Russia last week.
His counterpart, Sergei Lavrov, said there was no proof for Mr Johnson's claim.
While Mr Johnson said he wants to normalise relations with Russia, Moscow blames London for the poor state of relations.
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