In early 1960, Frank Morris (Clint Eastwood), a prisoner with an exceptional I.Q. who has absconded from other facilities, arrives at the maximum security prison on Alcatraz Island. Soon after arriving, he is summoned to the office of the warden (Patrick McGoohan), who curtly informs him that Alcatraz is unique within the U.S. prison system for its exceedingly high level of security and that no inmate has ever successfully escaped. During the conversation, the warden fails to notice Morris steal one of the nail clippers on the desk.
Over the next several days, Morris makes acquaintances with a few inmates, including the eccentric Litmus (Frank Ronzio), who is fond of desserts; English (Paul Benjamin), a black inmate serving two life sentences for killing two white men in self-defense; and the elderly Doc (Roberts Blossom), who paints portraits and once grew chrysanthemums at Alcatraz. Morris also makes an enemy of a rapist called Wolf (Bruce M. Fischer), who tries to harass him in the showers and later attacks him in the prison yard with a knife; both men are imprisoned in isolation in the hole. Morris is released while Wolf stays. When the warden discovers that Doc has painted an ungainly caricature of him, as well as other policemen on the island itself, he permanently removes Doc’s painting privileges; in response, a depressed Doc hacks off his own fingers with a hatchet from the prison workshop and is led away. Later, Morris encounters bank robber brothers John and Clarence Anglin (Fred Ward and Jack Thibeau), who are his old friends from another prison sentence, and he makes the acquaintance of prisoner Charley Butts (Larry Hankin). Morris notices that the concrete around the grille in his cell is weak and can be chipped away, which evolves into an escape plan. Over the next few months, Morris, the Anglins, and Butts dig through the walls of their cells with spoons (which have been soldered into makeshift shovels), make papier-mâché dummies to act as decoys, and construct a raft out of raincoats.
Later, during mealtime, Morris places a chrysanthemum at the table in honor of Doc, but the warden stops by and crushes it, causing a provoked and angry Litmus to suffer a heart attack. Later, the warden requests a shakedown of Morris’s cell and finds nothing unusual. However, he issues orders for Morris to be relocated to a different cell as soon as possible. Wolf has been released from solitary confinement and prepares to attack Morris again, but English is able to intercept him, with English implying that his gang will beat up Wolf. That night, the inmates decide to leave. Morris, the Anglins and Butts plan to meet in the passageway and escape. Butts loses his nerve and fails to rendezvous with them. He later changes his mind but is too late and returns to his cell where he sulks over his missed opportunity. Carrying the flotation gear, Morris and the Anglins access the roof and avoid the searchlights. From there, they scramble down the side of the building into the prison yard, climb over a barbed-wire fence, and make their way to the shoreline of the island, where they inflate the raft. The three men depart from Alcatraz, partially submerged in the water, clinging to the raft and using their legs as the primary propelling force.
The following morning, the escape is discovered and a massive manhunt ensues. Shreds of raincoat material, including personal effects of the men are found floating in the bay. While searching on Angel Island, the warden stubbornly insists that the men’s personal effects were important, and the men would have drowned before leaving them behind. However, another guard has his doubts, suggesting that the convicts got rid of them on the pretense that they drowned. The warden is informed by his aide that he has been summoned to go to Washington to face his superiors, with the prospect of being forced to accept early retirement for having failed to prevent the breakout. On a rock, the warden finds a chrysanthemum and is told by his aide that none grow on Angel Island. The film closes with an inserted text narrative telling that the fugitives were never found, and that Alcatraz was closed less than a year later.