Vladimir Putin has thanked Tony Blair for the UK's help in rescuing the stranded submariners.
Mr Putin expressed thanks to the UK for saving the submariners
The Russian president described the British contribution to the success of the operation as "of utmost value to Russia".
Further congratulations were added by Deputy Prime Minister John Prescott.
The 29-strong rescue team returned to Prestwick Airport on Tuesday. They freed the trapped submarine with a remote-controlled robot, Scorpio 45.
The Russian crewmen were trapped in their mini-submarine for three days after it got caught in fishing nets at the bottom of the Pacific Ocean.
The submarine crew had only six hours of oxygen left when they reached the surface.
The team was met at the airport by Armed Forces Minister Adam Ingram.
He said: "This was a magnificent achievement. I think we are all very relieved that the Russian crew are safe and well. That is down to the qualities of the crew that went out to rescue them... they've achieved something truly remarkable."
Stuart Gold, the team leader who operated Scorpio, said: "When the thing shot to the surface, it was something else.
"I must admit, I had a tear in my eye too.
"Then a day later, to see pictures of family and loved ones waiting by the phone to hear words that their fathers and brothers were safe was something I'll never forget."
'I was scared'
A British Ministry of Defence (MoD) spokesman confirmed all men aboard the AS-28 mini-submarine were in a satisfactory condition in hospital.
Scorpio 45 spent five hours using its cameras and robot arm to free the submarine, trapped 190m (620ft) down off the Kamchatka peninsula.
Commander Ian Riches, the Naval officer who headed the rescue team said he was "elated" at saving the men.
He said he had told the media before the mission that he was "optimistic".
"I lied," he said.
"I was scared. But I shouldn't have been because I have a team of professional men who worked damned hard to ensure that the guys in that Russian submarine got out alive."
He added: "It would be wrong of me to say that grown men don't cry because I can assure you, a lot of grown men cried that day."
Mr Ingram added: "I just want to pay a very big tribute to all of the people who have been involved in this mission.
"Particularly the crew themselves, who have carried out what was undoubtedly a difficult and technical job.
"It could only be achieved because of the training they undertook, because of their professionalism, and because of their commitment and dedication to the task."
The team in charge of the robot submersible, managed and operated for the Royal Navy by contractors Rumic Ltd, based in Renfrew, near Glasgow, are constantly on alert to fly to submarine emergencies anywhere in the world.