Popeye, a strong sailor, arrives at the small coastal town of Sweethaven while searching for his missing father. He rents a room at the Oyl family’s boarding house where the Oyls plan to have their daughter Olive become engaged to Captain Bluto, a powerful, perpetually angry bully who runs the town in the name of the mysterious Commodore. However, on the night of the engagement party, Olive sneaks out after discovering that the only attribute she can report for her bullying fiancé is size. She encounters Popeye, who failed to fit in with the townsfolk at the party. The two eventually come across an abandoned baby in a basket. Popeye and Olive adopt the child, naming him Swee’Pea after the town Sweethaven, and the two return to the Oyls’ home. Bluto finds out about this encounter, which results in him declaring heavy taxation for the Oyls’ property and possessions out of rage. A greedy taxman follows up on Bluto’s demand, but Popeye helps the Oyls’ financial situation by winning a hefty prize from defeating a boxer named Oxblood Oxheart.
The next day, Popeye discovers that Swee’Pea can predict the future by whistling when he hears the correct answer to a question. Wimpy, the constantly hungry local mooch and a petty gambler, also notices this and asks Popeye and Olive to take Swee’Pea out for a walk. However, he actually takes him to the “horse races” (actually a mechanical carnival horse game) and wins two games. Hearing of this, Olive and her whole family decide to get in on the action and use Swee’Pea to win, but an outraged Popeye finds out and takes Swee’Pea away.
Later, after Popeye throws the taxman into the sea (simultaneously earning the respect of the town), Wimpy then kidnaps the child at Bluto’s orders. Wimpy informs Popeye about the kidnapping, after being threatened by Olive. Popeye goes to the Commodore’s ship, where he finds out that the Commodore, who has actually been held a prisoner by Bluto all this time, is indeed Popeye’s father, Poopdeck Pappy, who accepts that Popeye is his son after exposing Popeye’s hatred of Spinach. Meanwhile, Bluto kidnaps Olive and sets sail with her and Swee’Pea to find buried treasure promised by Pappy. Popeye, Pappy, Wimpy and the Oyl family board Pappy’s ship to chase Bluto to Scab Island, a desolate island in the middle of the ocean.
Popeye catches up to Bluto and fights him, but despite his determination, Popeye is overpowered. During the fight, Pappy recovers his treasure and opens the chest to reveal a collection of personal sentimental items from Popeye’s infancy, including a few cans of spinach. A gigantic octopus awakens and attacks Olive from underwater (after Pappy saves Swee’Pea from a similar fate). With Popeye in a choke hold, Pappy throws him a can of spinach; Bluto, recognizing Popeye’s dislike for spinach, force-feeds him the can before throwing him into the water. The spinach revitalizes Popeye and boosts his strength, helping him to defeat both Bluto and the giant octopus. Popeye celebrates his victory and his new-found appreciation of spinach while Bluto swims off, having literally turned yellow.