The pair had been caught up in fighting in Libya and were among more than 100 British citizens taken to safety.
Photographs released by Ministry of Defence officials at the time showed the group being brought on board the Navy vessel.
A Whitehall source said: "For this man to have committed such an atrocity on UK soil after we rescued him from Libya was an act of utter betrayal."
The revelation will enrage families who lost loved ones in Abedi's despicable attack. It is also likely to raise fresh fears over possible intelligence failures.
Abedi was known to the security services and was being monitored at the time of his trip to Libya. However, just one month prior to his rescue, MI5 closed his case as a result of mistaken identity.
The presence of the Abedi brothers among the 110 evacuees from Libya in 2014 was confirmed by family friends in Libya. One said: "They were sent together by the Royal Navy to Malta."
Sources in London also confirmed Salman was on HMS Enterprise.
After being dropped off in Malta, Salman and his 21-year-old brother – the British-born sons of Libyan migrants – flew back to Manchester where they were living at the time.
Salman, who was on a gap year from Manchester College, went on to study business management at Salford University, before dropping out and descending into a fanatical spiral that culminated in last year's suicide bombing at the age of 22.
Two people who knew Salman Abedi from his college days claim they called an anti-terrorism hotline about him.
The Abedi brothers shuttled back and forth between Manchester and Tripoli because their parents – Ramadan and Samia – had returned to Libya.
Ramadan is thought to have gone back in time for the 2011 revolution, allegedly fighting against the Gaddafi regime with the Libyan Islamic Fighting Group.
It is not certain whether the two brothers were with their father at the time of the revolution or instead in neighbouring Tunisia.
But they were on holiday in Libya in August 2014 when civil war fighting broke out and British officials offered to evacuate UK citizens.
The Royal Navy was tasked with picking them up, along with other British nationals, on a list provided to sailors.
By that time, the Foreign Office had already changed its official travel advice to warn Britons in Libya to 'leave immediately by commercial means' because of the fighting around Tripoli and the wider instability nationwide.
The advice said those unable to leave independently could seek 'assisted departure'.
Senior security sources stressed they did not believe Abedi had been radicalised at the time of the Royal Navy rescue. He later became brainwashed after watching bomb-making videos on Google-owned YouTube and terror material on other internet sites, they believe.
A senior source told the Mail: "He was a British citizen so it was our job to safeguard him. Salman was one of many people in that mix and we absolutely had to evacuate him.
"He was not a threat at the time and it was in a very different context."
Other sources have claimed Abedi was on the front line and was hospitalised fighting alongside jihadis in Ajdabiya, eastern Libya.
However, a family friend was adamant that Abedi was there for innocent purposes. The friend said: "Salman and Hashem were not involved in fighting at all and they had spent a lot of time with their mother in Tunisia."
Abedi was on a gap year from Manchester College and he later went on to study business management at Salford University.
At the time of the rescue, there were increasingly brutal battles between various militia groups fighting for control of Tripoli's airport and dozens were killed.
As a result the Foreign Office announced it was temporarily closing its embassy and other consular operations in Libya.
Amid fierce fighting UK citizens were taken in small boats from the port of Tripoli to HMS Enterprise, a survey ship on a routine deployment in the Mediterranean. There were joined by two Irish citizens and a German.
It is not known if the flights home to the UK were paid for by the British government.
A security source downplayed claims that Abedi was involved in fighting and said he could have just been visiting his family.
"When they leave the UK it becomes a lot more challenging to keep track of them, especially when they have family links in other countries," said the insider.
"But his visits were not necessarily for nefarious purposes. He did things largely alone. The internet played a large role when it came to his terror training."
A report into the handling of the Manchester bombing by David Anderson QC revealed that Abedi was first actively investigated in January 2014 – seven months before the rescue.
Police escort members of the public from the Manchester Arena after a suicide bomber killed 22 people. Photo / Getty
Published in December last year, the report said he was investigated because it was "thought that he might have been an individual who had been seen acting suspiciously with a subject of interest" to counter-terrorism police.
Abedi did know the suspect but turned out not to have been the individual seen with him and his record was closed in July 2014. He was classed as a low residual risk.
He came to the attention of the authorities again in October 2015 because of his supposed contact with an Islamic State figure in Libya.
The report found that this allegation was wrong and his file was closed once more.
Abedi attended Burnage Academy for Boys in Manchester between 2009 and 2011 before going to Manchester College until 2013. Two people who knew him from his college days claim they called an anti-terrorism hotline to warn police about his extremist views.
A community support worker, who did not wish to be named, told the BBC they had informed the authorities after Abedi publicly said "he was supporting terrorism" and that "being a suicide bomber is OK".
Greater Manchester Police found no record of the phone calls, however.
The Anderson review concluded that the investigative actions taken in relation to Abedi, and the subsequent decision to shut his file, were sound on the basis of the information available at the time.
At the time of the rescue in 2014, the MoD released a statement from HMS Enterprise's commanding officer, Cdr Mark Varta, saying: "This is a period of uncertainty for UK citizens based in Libya but we have been proud to play our part in enabling their move to safety."
The then defence secretary, Sir Michael Fallon, said: "I thank the crew of HMS Enterprise for their support and professionalism in carrying out this important task."
The 90-metre-long survey ship, normally based in Plymouth, had been eight weeks into an 18-month mission carrying out surveys in the area.
Last night, a Government spokesman said: "During the deteriorating security situation in Libya in 2014, Border Force officials were deployed to assist with the evacuation of British nationals and their dependents."
