Quoting banking officials, the agency said the gang behind the raid used stolen credentials to make requests to transfer cash look legitimate.
If all the requests had gone unchallenged, the gang would have got away with about $1bn.
However, the transfers were stopped when the volume of requests raised suspicions at other banks.
The computer hackers behind the breach are believed to have got away with about $80m - one of the largest known bank robberies in history.
To commit the attack, which took place about a month ago, the gang spent time studying the internal processes of Bangladesh's central bank so they could convincingly pose as officials when requesting the transfers.
However, said Reuters, the sheer number of transactions and a spelling mistake helped alert bank staff to the theft.
The spelling mistake in the name of one recipient of funds led Deutsche Bank, which was helping route the cash, to ask for clarification from the central bank, which then stopped the transaction.
Around the same time, the Federal Reserve Bank of New York had alerted the Bangladesh central bank to a series of suspicious requests to transfer money. The requests are believed to have been flagged because they were to private accounts rather than other banks and because there were so many of them.
If all the transfer requests had been fulfilled, the thieves would have got away with about $950m.
The $80m the hackers did steal ended up in accounts in Sri Lanka and the Philippines, said Reuters. The central bank has started proceedings to recover the cash.
Bangladesh's government has publicly blamed the New York Fed for not spotting the suspicious transactions earlier. Earlier this week Bangladesh's Finance Minister Abul Maal Abdul Muhith said it might launch legal action against the US body to help recover the money.
"The Fed must take responsibility," he said.
The New York Fed said its system had not been breached by the hackers and added that it was working with Bangladesh's central bank to investigate what happened.
Security and forensics firm Mandiant is also helping the Bangladeshi investigation into the theft.

Haki Africa said the figure could be higher but some families were reluctant to come forward for fear of victimisation by state security agents.
The police oversight body said 52 officers were on trial in connection with such killings countrywide.
The coastal region has suffered several terror attacks in recent years.
The Somali Islamist militant al-Shabab group has been targeting Kenya over the deployment of Kenyan forces to Somalia.
Most of the victims were youths including returnees from Somalia who had surrendered after the government promised them amnesty, Haki Africa's report said.
Others were allegedly killed as a result of excessive use of force during police crackdowns on protests and other religious groups that had been accused of radicalising young people.
Saada Suleiman, who attended the report's launch in the capital, Nairobi, said her husband had been missing for two years since his arrest in a raid on Masjid Musa mosque in Mombasa known for its radical sermons.
Haki Africa Executive Director Hussein Khalid said some of those killed were terror suspects, including several radical Muslim clerics, but this was no excuse for the state to execute them without due process.
The BBC's Abdinoor Aden in the capital, Nairobi, says officers from the anti-terror unit have been heavily implicated in the report.
Murshid Nassir, who heads the National Police Service Commission which vets officers, told the BBC that so-called "death squads" did not exist within the force and extrajudicial killings were not condoned.
While the chairman of the Independent Policing Oversight Authority promised to take action against officers found guilty of engaging in the killings.
Macharia Njeru said 52 officers had been charged and more than 300 officers were being probed for gross human rights violations.
Cyberthieves who targeted Bangladesh's central bank tried to get away with $1bn (Â£700m), reports Reuters.
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Kenya's anti-terror police have carried out at least 81 extrajudicial killings in country's mainly Muslim coastal region since 2012, a rights group says.