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While browsing in an outdoor market with his pet dog Snowy, Tintin, a young journalist, purchases a miniature model of a ship known as the Unicorn, but is accosted by two men, who both separately 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 stray cat; a metal tube slips out of the ship’s mast and rolls under Tintin’s furniture. Meanwhile, bumbling police detectives Thomson and Thompson are on the trail of pickpocket Aristides Silk, whom Snowy had seen in the market.

After visiting library to uncover the history surrounding the Unicorn, Tintin returns to find the Unicorn has been stolen, and suspects Sakharine of stealing it. He heads to Marlinspike Hall with Snowy, where he accuses Sakharine 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 metal tube by Snowy. Tintin finds a parchment scroll inside the tube. However, they are interrupted by the arrival of Barnaby, the Interpol officer who had tried to buy the model from Tintin, who is then assassinated while attempting to recover the Unicorn.

Tintin places the scroll in his wallet, but is then pick-pocketed by Silk the next morning and soon afterwards abducted by accomplices of Sakharine on the SS Karaboudjan. Snowy chases after them, boards the ship and reunites with Tintin. Tintin learns that Sakharine formed an alliance with the ship’s staff and led a mutiny to take control and imprison the captain. On board, Tintin encounters the ship’s captain, Archibald Haddock, who is a perpetually drunk alcoholic and has forgotten most of his past including his ancestor’s story. Tintin, Haddock and Snowy outrun the crew and escape from the Karaboudjan in a lifeboat, but not before Tintin radios the name of the ship’s destination, Bagghar. 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 alcohol to put out a fire he started in the boat, causing an explosion that leaves the trio stranded on top of the capsized boat. The trio find the seaplane attacking them, shoot it down and hijack the seaplane and use it to fly towards the fictitious Moroccan port of Bagghar, but it crashes in a desert due to it running out of fuel and flying in a thunderstorm. While walking in the desert, 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. before becoming sober. However, they are later saved by European soldiers. Meanwhile, Thomson and Thompson find and arrest Silk, who claims to be a kleptomaniac and not a thief, and find Tintin’s wallet with the contents still inside.

At the desert campbase, Haddock, while drunk, remembers that Sir Haddock’s ship was ambushed by a pirate ship led by Red Rackham, who is later revealed to be Sakharine’s ancestor. Sir Francis surrendered on the condition that his crew be allowed to live, but since the pirates killed his entire 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 the location of the sunken Unicorn and its treasure.

The third model is revealed to be in Bagghar, possessed by Omar ben Salaad. Sakharine causes a distraction in a Bianca Castafiore concert, which results in his falcon 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 Antwerp 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 fight using 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 Sir Francis’ sunken treasure.