"The ship's company dealt with the incident quickly and professionally and, as a result, there is only superficial damage to the forward compartment," it said.
The submarine - designed to attack enemy shipping - does not carry any nuclear-armed missiles.
Sailor airlifted to safety
Following the accident, the submarine surfaced through the Arctic ice and the injured crewman was flown to a hospital in Anchorage, Alaska.
The Tireless was operating with the USS Alexandria, a US submarine, in a joint operation to test submarine operability and tactical development in Arctic waters.
Since 1986, every Arctic tactical exercise has involved both US Navy and Royal Navy submarines.
Andy Price, a spokesman for the Royal Navy, said the submarine, while still fully functional, would be evaluated over the next 12 hours to determine whether it will continue to be part of the joint exercises or return to the United Kingdom.
Two weeks of exercises are scheduled to end March 30.