Early indications were that the accident on board HMS Tireless involved a piece of air purification equipment to the bow (front) of the submarine, but the vessel was "never in any danger," the MoD said.
"The MoD can confirm that at (0430 GMT) this morning there was an accident onboard a Trafalgar Class submarine on exercise in the Arctic," it said in a statement.
"The submarine, HMS Tireless, was never in any danger, its nuclear reactor was unaffected, it quickly surfaced and is completely safe."
The nuclear-powered hunter-killer submarine, which was taking part in a joint exercise with the US Navy, does not carry nuclear missiles.
The MoD said the family of the two crew members had been informed while the third who was injured was airlifted to a US military hospital.
His injuries are not thought to be life-threatening and he is expected to make a full recovery, it added.
"At this early stage, it is thought that the accident involved a piece of air purification equipment in the forward section of the submarine," the statement read.
"The ship's company dealt with the incident quickly and professionally and, as a result, there is only superficial damage to the forward compartment...
"The crew are trained in surfacing quickly through the ice, and did so in exemplary fashion."
Admiral Sir James Burnell-Nugent, commander-in-chief (fleet) of the Royal Navy and vice-admiral of the United Kingdom, voiced his regrets but added his praise for the crew.
"I very much regret that this incident has occurred and my thoughts go out to the family and friends of the men who have lost their lives," he said in a statement.
"I also wish to pay tribute to the crew of HMS Tireless that this incident has been dealt with and contained so professionally."
Air purification equipment is fitted to all Trafalgar Class submarines, of which Tireless is one of seven in the Royal Navy.
The MoD said the equipment has a 100 percent safety record to date, but as a precaution its use on other submarines has been restricted until safety checks can be carried out.
Tireless, which is based in Davenport, in Plymouth, south-west England, was launched in 1984, but the piece of air-purification machinery thought to have failed was fitted as part of an update in 2001.
In January this year, a US Navy nuclear-powered submarine USS Newport News collided with a Japanese tanker in the Strait of Hormuz in the Arabian Sea, one of the world's busiest seaways, but no injuries were reported.
Two crew members from the USS Minneapolis-St Paul died after being hit by a wave while they were standing on the hull of their vessel as it left port in south-west England last December.
In September last year, two submariners died and one was injured in a fire on board the Russian nuclear submarine the Saint Daniel of Moscow in the Barents Sea near Norway.
In February 2003, it emerged that an Australian submarine, the HMAS Dechaineux, was just 20 seconds from sinking to the bottom of the Indian Ocean with 55 sailors on board, prompting safety fears.
The biggest submarine catastrophe was the sinking of Russia's Kursk in August 2000, also in the Barents Sea, in which 118 crew died.

Copyright AFP (Agence France-Presse), 2007