The captured sailors have been shown on Iranian television
Iranian TV footage
It also calls on Tehran to allow the UK consular access to the personnel, but stops short of "deploring" Iran's action, as requested by the UK.
Iran, which the UK later confirmed had sent it a note, said it was not helpful to try to engage third parties.
Meanwhile, an Iranian TV channel says it is to broadcast a "confession" by one of the detained navy personnel.
Official TV channel Al-Alam did not say what the confession would contain or the identity of the sailor involved.
The Foreign Office said of the note, sent to the British embassy: "We can confirm that, as reported in the Iranian media, the Iranian government has sent a formal note to the British embassy.
Confidential
"Such exchanges are always confidential, so we cannot divulge any details, but we are giving the message serious consideration and will soon respond formally to the Iranian government."
Sir Menzies Campbell, the Liberal Democrat leader, told BBC Radio 4's Today programme that the note may contain "matters of substance" which may be the beginning of a "proper exchange" and could "offer the possibility of a solution".
This case can and should be settled through bilateral channels.
Iran's UN mission
Letter urges Iraq withdrawal
On a visit to British troops in Afghanistan on Friday, Chancellor Gordon Brown welcomed the UN statement.
He said it was calling definitively for the navy crew's release.
"That's the unanimous view of the international community."
Mr Brown described the treatment of captured Leading Seaman Faye Turney, 26, an interview with whom has been shown on Iranian TV, as "cruel, callous, inhuman and unacceptable".
Responding to the UN statement, David Cameron, the Conservative leader, said: "Really there's only one right outcome to this which is for Iran to release those people that they took captive, that they should be released without any further ado. It's as clear and as simple as that."
The British ambassador to the UN, Sir Emyr Jones Parry, said the UN's statement was a "good outcome" for the UK.
Sir Emyr added that it sent the "right message" to Iran to allow access to the sailors and marines, and secure their prompt release.
UK VERSION OF EVENTS 1 Crew boards merchant ship 1.7NM inside Iraqi waters
2 HMS Cornwall was south-east of this, and inside Iraqi waters
3 Iran tells UK that merchant ship was at a different point, still within Iraqi waters
4 After UK points this out, Iran provides alternative position, now within Iranian waters
UK version in more detail
The UK failed to win support for a stronger statement deploring Iran's actions, following opposition led by Russia.
Iran's UN mission said in a statement: "This case can and should be settled through bilateral channels.
"The British government's attempt to engage third parties, including the Security Council, with this case is not helpful."
Satellite data
The UN Security Council statement was agreed following more than four hours of negotiations.
In full, it said: "Members of the Security Council expressed grave concern at the capture by the Revolutionary Guard and the continuing detention by the government of Iran of 15 United Kingdom naval personnel and appealed to the government of Iran to allow consular access in terms of the relevant international laws.
"Members of the Security Council support calls, including by the secretary general in his March 29 meeting with the Iranian foreign minister, for an early resolution of this problem, including the release of the 15 UK personnel."
The Britons, based on HMS Cornwall, were seized a week ago by Iranian Revolutionary Guards as they returned from searching a vessel in the northern Gulf.
Iran says they had strayed into Iranian territorial waters, a claim which the UK has denied.
Earlier this week, the Royal Navy produced satellite data it said proved its case.
Second 'letter'
Meanwhile, Iran has released a second letter, apparently written by Leading Seamen Turney, which urges the UK to withdraw from Iraq.
The letter, which was released by the Iranian embassy in London and addressed to British MPs, states: "Isn't it time for us to start withdrawing our forces from Iraq and let them determine their own future?"
It also says that "even through our wrongdoing" the Iranians have "treated us well and humanely".
On Thursday, Ali Larijani, Iran's chief nuclear negotiator, said the release of Leading Seamen Turney had been delayed, and spoke about the UK's "incorrect attitude".
The United Nations Security Council has issued a statement calling for a "early resolution" to the rising tensions between United Kingdom and Iran by releasing all 15 British sailors and marines who were detained by Iranian forces on March 23.

"Members of the Security Council expressed grave concern at the capture by the Revolutionary Guard and the continuing detention by the government of Iran of 15 United Kingdom naval personnel and appealed to the government of Iran to allow consular access in terms of the relevant international laws," said the U.N. Security Council in a statement.

"Members of the Security Council support calls, including by the secretary general in his March 29 meeting with the Iranian foreign minister, for an early resolution of this problem including the release of the 15 UK personnel," added the statement.
United Nations Security Council.
All 15 members of the Council agreed to the wording of the statement and released it after nearly four hours of debate, but Russia and South Africa opposed to having the statement or the Council's recommendations be "too harsh" or to put the blame fully on the Iranian government.

Iran released a statement shortly after the Security Council met saying that getting the U.N. involved, "will not be helpful."

"This case can and should be settled through bilateral channels. The British government's attempt to engage third parties, including the Security Council, with this case is not helpful," said a statement released by Iran's U.N Mission.

The sailors and marines from the frigate  had been inspecting, in accordance with UN Security Council Resolution 1723, a ship that was believed to be smuggling cars into Iraq, though it was subsequently cleared after inspection when Iranian gunboats surrounded the sailors and arrested them at gunpoint.

On Thursday, Iran stated that they are "suspending" the release of the 15 soldiers, including  a female soldier, 26, who was supposed to be released "very soon."