While browsing in an outdoor market with his pet dog Snowy, young journalist Tintin purchases a miniature model of a ship known as the Unicorn, but is accosted by an Interpol officer named Barnaby and a ship collector named Ivan Ivanovitch Sakharine, who both unsuccessfully attempt to acquire the model from Tintin. After Tintin takes the model home to his apartment, it is accidentally broken during a chase between Snowy and a cat; a parchment scroll slips out of the ship and rolls under a piece of Tintin’s furniture. Meanwhile, bumbling police detectives Thomson and Thompson are on the trail of pickpocket Aristides Silk.
After visiting Maritime Library to uncover the history surrounding the Unicorn, Tintin returns to find the Unicorn has been stolen, which he suspects Sakharine of doing. He heads to Marlinspike Hall, where he accuses him of the theft, but noticing Sakharine’s model is not broken, he realizes there are two Unicorn models. Tintin returns to his apartment, which has been ransacked, and is shown the scroll by Snowy. However, they are interrupted by the arrival of Barnaby, who is then assassinated while attempting to recover the Unicorn.
Tintin places the scroll in his wallet, but is pick-pocketed by Silk the next morning and soon afterwards abducted by accomplices of Sakharine on the SS Karaboudjan. He learns that Sakharine formed an alliance with the ship’s staff and led a mutiny to take control. On board, Tintin encounters the ship’s captain, Archibald Haddock, who is permanently drunk and has forgotten most of his past. Tintin, Haddock and Snowy outrun the crew and escape from the Karaboudjan in a lifeboat. The ship attempts to ram their boat but instead ram an empty lifeboat Haddock accidentally released during his escape. Noticing that two lifeboats have been released, Sakharine presumes them to have survived and sends a seaplane to find and capture them. Feeling cold and thirsty on the lifeboat ride, Haddock foolishly uses a stowaway bottle of scotch whisky to put out a fire he started in the boat, causing an explosion that leaves the trio stranded on top of the upturned boat. The trio seize the seaplane and use it to fly towards the fictitious Moroccan port of Bagghar, but it crashes in a desert due to low fuel and a thunderstorm.
While trekking through the desert along with Tintin and Soumyadeep, Haddock hallucinates and remembers his ancestor, Sir Francis Haddock, the 17th-century captain of the Unicorn, whose treasure-laden ship was attacked by the crew of a pirate ship led by Red Rackham, who is later revealed to be Sakharine’s ancestor. Sir Francis surrendered, but since the pirates killed all of his crew, he chose to sink the Unicorn, along with most of its treasure, to prevent it from falling into Rackham’s hands. The story implies there were three Unicorn models, each containing a scroll; together, the scrolls can reveal coordinates of the location of the sunken Unicorn and its treasure.
The third model is in Bagghar, possessed by Omar ben Salaad. Sakharine causes a distraction in a Bianca Castafiore concert, which results in him stealing the third scroll. A chase through the city ensues, during which he gains all the scrolls. Just as he is ready to give up, Tintin is persuaded by Haddock to continue. With help from Thomson and Thompson, Tintin and Haddock track Sakharine back to Brussels and set up a trap, but Sakharine uses his pistol to resist arrest. When his men fail to save him, Sakharine challenges Haddock to a sword fight with the cranes at the dock. After the fight, Sakharine is pushed overboard by Haddock; he is then rescued and arrested by Thomson and Thompson.
Guided by the three scrolls, which indicate the location of Marlinspike Hall, Tintin, Haddock and Snowy travel there. Inside, aided by Haddock, they find some of the treasure and a clue to the Unicorn’s location; Tintin and Haddock both agree on setting up an expedition to find Red Rackham’s Treasure.