As shown in the film, they spent months chipping through the walls of their cells with sharpened metal spoons into an unguarded corridor that led out of the jail, popularly known as "the Rock".
They made an inflatable raft from 50 stolen raincoats, and fashioned fake papier-mache heads using toilet paper and hair from the prison barbershop, which they left on their pillows to fool guards that night.
An FBI investigation concluded that Morris and the Anglin brothers set off from the north side of Alcatraz Island sometime between 8pm and 2am, braving the powerful and unpredictable currents of the bay.
Using a computer model to simulate tides on the night, scientists from Delft University and the Dutch research institute Deltares have found that the trio could have made landfall only if they had left the island between 11pm and midnight.
The researchers simulated the journeys of 50 virtual rafts for every 30 minutes of the night in question, from several possible points of departure on the island.
The study found that if they set off before 11pm, they would have been swept out under the Golden Gate Bridge into the Pacific, where they would almost certainly have died.
If they left after 1am, the tide would have turned, taking them back into the bay towards Oakland or Berkeley. "In both cases they would have spent so much time in the water, they probably would have died of hypothermia, or they would have been picked up by the police because sunrise was at 6am," said researcher Dr Rolf Hut.
But if they left Alcatraz between 11pm and midnight, pointed north and paddled furiously - they had oars fashioned from musical instruments - they would have been dragged towards the Golden Gate only to find the tide turning at the last moment, possibly helping them to make landfall on the north side of the bridge.
The study is being presented in San Francisco this week, at a meeting of the American Geophysical Union.
The men were never found and were eventually presumed to have died. The Dutch study suggested that, had they died, the remains of their raft would probably have washed up on Angel Island, another island in the bay - which is where investigators found a paddle and personal items in 1962.
Alcatraz
• Originally a military base, the island became a military prison in 1861, during the American Civil War.
• In 1933 it became a federal penitentiary, and over the next 30 years housed many infamous criminals, including Al Capone, Mickey Cohen and James "Whitey" Bulger.
• 36 men tried to escape in 14 incidents. Six were shot and killed, 23 were caught, two drowned and five were listed as presumed drowned.
• Today it is a tourist attraction.